<PAGE>   1
 
================================================================================
 
                       SECURITIES AND EXCHANGE COMMISSION
                             Washington, D.C. 20549

                             ---------------------
 
                                   FORM 10-Q
 
<TABLE>
<S>       <C>                                                          <C>
[X]             QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15(d)
                     OF THE SECURITIES EXCHANGE ACT OF 1934

                 FOR THE QUARTERLY PERIOD ENDED MARCH 31, 1998

                                       OR

[ ]            TRANSITION REPORT PURSUANT TO SECTION 13 OR 15(d)
                     OF THE SECURITIES EXCHANGE ACT OF 1934
</TABLE>
 
             FOR THE TRANSITION PERIOD FROM ________ TO ________ .

                         COMMISSION FILE NUMBER 0-8565
 
                             MARINE PETROLEUM TRUST
             (Exact name of registrant as specified in its charter)
 
<TABLE>
<S>                                            <C>
                    TEXAS                                        75-6008017
         (State or other jurisdiction                         (I.R.S. Employer
      of incorporation or organization)                     Identification No.)

          NATIONSBANK OF TEXAS, N.A.                             75283-0241
        P.O. BOX 830241, DALLAS, TEXAS                           (Zip Code)
   (Address of principal executive offices)
</TABLE>
 
       Registrant's telephone number, including area code (800) 985-0794
 
                                      None
              (Former name, former address and former fiscal year
                         if changed since last report)
                             ---------------------
 
     Indicate by check mark whether the registrant (1) has filed all reports
required to be filed by Section 13 or 15(d) of the Securities Exchange Act of
1934 during the preceding 12 months (or for such shorter period that the
registrant was required to file such reports), and (2) has been subject to such
filing requirements for the past 90 days.
 
                             Yes   X       No 
                                 ----         ----
     Indicate number of units of beneficial interest outstanding as of the last
practicable date.
 
<TABLE>
<CAPTION>
         Title of Each Class of Units              Number of Units of Beneficial Interest
            of Beneficial Interest                             March 31, 1998
         ----------------------------              --------------------------------------
<S>                                            <C>
         UNITS OF BENEFICIAL INTEREST                            2,000,000
</TABLE>
 
================================================================================
<PAGE>   2
 
PART I. FINANCIAL INFORMATION
 
                     MARINE PETROLEUM TRUST AND SUBSIDIARY
 
                     CONDENSED CONSOLIDATED BALANCE SHEETS
                        MARCH 31, 1998 AND JUNE 30, 1997
                                  (UNAUDITED)
 
                                     ASSETS
 
<TABLE>
<CAPTION>
                                                               MARCH 31      JUNE 30
                                                                 1998          1997
                                                              ----------    ----------
<S>                                                           <C>           <C>
Current Assets:
  Cash and cash equivalents.................................  $2,088,599    $1,759,718
  Oil and gas royalties receivable..........................     264,475       835,230
  Receivable from affiliate.................................     110,339        64,580
                                                              ----------    ----------
          Total current assets..............................   2,463,413     2,659,528
                                                              ----------    ----------
Investment in affiliate.....................................     337,799       285,478
Office equipment, at cost less accumulated depreciation.....         555           272
Producing oil and gas properties............................           7             7
                                                              ----------    ----------
                                                              $2,801,774    $2,945,285
                                                              ==========    ==========
 
                             LIABILITIES AND TRUST EQUITY
 
Current Liabilities:
  Accounts payable..........................................  $1,010,250    $  895,724
  Income taxes payable......................................       4,858         4,758
                                                              ----------    ----------
          Total current liabilities.........................   1,015,108       900,482
Trust Equity:
  Corpus -- authorized 2,000,000 units of beneficial
     interest, issued 2,000,000 units at nominal value......           8             8
  Undistributed income......................................   1,786,658     2,044,795
                                                              ----------    ----------
          Total trust equity................................   1,786,666     2,044,803
                                                              ----------    ----------
                                                              $2,801,774    $2,945,285
                                                              ==========    ==========
</TABLE>
 
                                        2
<PAGE>   3
 
                     MARINE PETROLEUM TRUST AND SUBSIDIARY
 
                  CONDENSED CONSOLIDATED STATEMENTS OF INCOME
       FOR THE THREE MONTHS AND NINE MONTHS ENDED MARCH 31, 1998 AND 1997
                                  (UNAUDITED)
 
<TABLE>
<CAPTION>
                                                   THREE MONTHS              NINE MONTHS
                                               --------------------    ------------------------
                                                 1998        1997         1998          1997
                                               --------    --------    ----------    ----------
<S>                                            <C>         <C>         <C>           <C>
Income:
  Oil and gas royalties......................  $562,845    $882,772    $1,996,217    $2,829,543
  Equity in earnings of affiliate............   134,457      82,690       384,689       233,386
  Interest income of subsidiary..............    26,937      24,898        74,837        73,776
                                               --------    --------    ----------    ----------
                                               $724,239    $990,360    $2,455,743    $3,136,705
                                               --------    --------    ----------    ----------
Expenses:
  General and administrative.................    38,230      36,520       111,070       112,259
                                               --------    --------    ----------    ----------
  Income before Federal income taxes.........   686,009     953,840     2,344,673     3,024,446
  Federal income taxes of subsidiary.........       400        (200)        6,100         5,300
                                               --------    --------    ----------    ----------
          Net income.........................  $685,609    $954,040    $2,338,573    $3,019,146
                                               ========    ========    ==========    ==========
Net income per unit..........................  $   0.34    $   0.48    $     1.17    $     1.51
                                               ========    ========    ==========    ==========
Distributions per unit.......................  $   0.51    $   0.52    $     1.30    $     1.51
                                               ========    ========    ==========    ==========
</TABLE>
 
                                        3
<PAGE>   4
 
                     MARINE PETROLEUM TRUST AND SUBSIDIARY
 
                CONDENSED CONSOLIDATED STATEMENTS OF CASH FLOWS
               FOR THE NINE MONTHS ENDED MARCH 31, 1998 AND 1997
                                  (UNAUDITED)
 
<TABLE>
<CAPTION>
                                                                 1998           1997
                                                              -----------    -----------
<S>                                                           <C>            <C>
Cash flows from operating activities:
  Net income................................................  $ 2,338,573    $ 3,019,146
  Adjustments to reconcile net income to net cash provided
     by operating activities:
     Depreciation...........................................        1,936            524
     Changes in assets and liabilities:
       Oil and gas royalties receivable.....................      570,755        204,544
       Receivable from affiliate............................      (45,759)        20,753
       Accounts payable.....................................      114,526             --
       Employment tax payable...............................           --         (2,459)
       Income taxes payable.................................          100         (2,312)
                                                              -----------    -----------
          Net cash provided by operating activities.........    2,980,131      3,240,196
                                                              -----------    -----------
Cash flows provided by investing activities:
       Purchase of office equipment.........................       (2,219)            --
       Decrease (increase) in undistributed earnings of
        affiliate...........................................      (52,321)         6,565
Cash flows from financing activities -- distributions to
  unitholders...............................................   (2,596,710)    (3,012,162)
                                                              -----------    -----------
          Net increase (decrease) in cash and cash
             equivalents....................................      328,881        234,599
                                                              -----------    -----------
Cash and cash equivalents at beginning of period............    1,759,718      2,020,695
                                                              -----------    -----------
Cash and cash equivalents at end of period..................  $ 2,088,599    $ 2,255,294
                                                              ===========    ===========
</TABLE>
 
                                        4
<PAGE>   5
 
                     MARINE PETROLEUM TRUST AND SUBSIDIARY
 
              NOTES TO CONDENSED CONSOLIDATED FINANCIAL STATEMENTS
                                 MARCH 31, 1998
                                  (UNAUDITED)
 
ACCOUNTING POLICIES
 
     The financial statements include the financial statements of Marine
Petroleum Trust (the "Trust") and its wholly-owned subsidiary, are condensed,
and should be read in conjunction with the annual report for the fiscal year
ended June 30, 1997. The financial statements included herein are unaudited, but
in the opinion of management include all adjustments necessary for a fair
presentation of the results of operations for the periods indicated.
 
UNDISTRIBUTED INCOME
 
     Undistributed income on March 31, 1998 includes $1,053,569 applicable to
the Trust and $733,089 applicable to Marine Petroleum Corporation, the Trust's
wholly-owned subsidiary. Distributions to unitholders are dependent on the
volume and price of oil and gas sold by others and will fluctuate from quarter
to quarter.
 
ACCOUNTS PAYABLE
 
     Marine Petroleum Corporation has provided an account payable of $843,250 to
cover possible refunds that may be required upon redetermination of gas prices
for royalty payments in prior periods. During the most recent quarter, Chevron
requested repayment for certain royalties that Chevron believes Chevron
incorrectly paid to Marine Petroleum Corporation during 1995, 1996 and 1997.
Marine Petroleum Corporation currently is reviewing this matter and has
established a reserve of $167,000 in accounts payable to cover this potential
liability.
 
     During the most recent quarter, Marine Petroleum Corporation also
determined that $52,474 of the previously-established account payable for
possible royalty overpayments no longer was required. Accordingly, during the
most recent quarter, Marine Petroleum Corporation reduced accounts payable, and
increased net income, by such amount.
 
                                        5
<PAGE>   6
 
                    MANAGEMENT'S DISCUSSION AND ANALYSIS OF
                 FINANCIAL CONDITION AND RESULTS OF OPERATIONS
 
FINANCIAL CONDITION -- LIQUIDITY AND CAPITAL RESOURCES
 
     The Trust is a "royalty trust" with overriding royalty interests in oil and
gas leases in the Gulf of Mexico. The Trust's indenture (and the charter and
by-laws of its subsidiary) expressly prohibit the operation of any kind of trade
or business. All royalties received by the Trust, less administrative expenses,
are distributed quarterly to unitholders. Since the Trust's sole purpose is to
collect and distribute cash collected from royalties, there are no requirements
for capital.
 
GENERAL
 
     Net income for the nine months ended March 31, 1998 amounted to $1.17 per
unit which was down approximately 23% from the $1.51 realized for the nine
months ended March 31, 1997. Income from royalties declined approximately 29%
and the Trust's equity in the income of Tidelands Royalty Trust B increased
approximately 65% for the current nine months as compared to the comparable
period a year ago.
 
     Distributable income will vary from quarter to quarter. The amount of each
distribution is determined by the amount of cash available for distribution on
the date the distribution is determined. The amount of each distribution is
determined 10 days before the applicable record date.
 
     The following table compares quarterly net income and distributable income
per unit for the past five quarters.
 
<TABLE>
<CAPTION>
QUARTER                                                       NET INCOME   DISTRIBUTION
-------                                                       ----------   ------------
<S>                                                           <C>          <C>
March 31, 1997..............................................     $.48          $.52
June 30, 1997...............................................      .45           .46
September 30, 1997..........................................      .36           .44
December 31, 1997...........................................      .47           .35
March 31, 1998..............................................      .34           .51
</TABLE>
 
     The Trust's current fiscal year began on July 1, 1997. Since that date,
operators have completed 26 new development wells on leases in which the Trust
has an interest. Eight of the wells were not successful or were abandoned, while
18 were completed as either oil or gas producers. At the present time operators
have 11 development wells in process and 7 locations for development wells to be
drilled in the future. The Trust understands that approximately half of the
wells are side tracked out of existing wells that are no longer producing.
 
     The Trust's revenues are derived from the oil and gas production activities
of unrelated parties. The Trust's revenues and distributions fluctuate from
period to period based upon factors beyond The Trust's control, including
without limitation the number of productive wells drilled and maintained on
leases subject to the Trust's interest, the level of production over time from
such wells and the prices at which the oil and gas from such wells is sold. The
Trust believes that it will continue to have revenues sufficient to permit
distributions to be made to unitholders for the foreseeable future, although no
assurance can be made regarding the amounts thereof. The foregoing sentence is a
forward-looking statement. Factors that might
 
                                        6
<PAGE>   7
 
cause actual results to differ from expected results include reductions in
prices or demand for oil and gas, which might then lead to decreased production;
reductions in production due to depletion of existing wells or disruptions in
service, including as the result of storm damage to production facilities,
blowouts or other production accidents, and geological changes such as cratering
of productive formations; expiration or release of leases subject to the Trust's
interests; and the discontinuation by parties subject to the contract dated
April 30, 1951 between Tidelands' predecessors and Gulf Oil Corporation of their
efforts to obtain leases in the area that is subject to Tidelands' interests.
 
RESULTS OF OPERATIONS -- THREE MONTHS ENDED MARCH 31, 1998
 
     Net income for the quarter ended March 31, 1998 amounted to $685,609, which
was approximately 28% less than the $954,040 realized in the comparable period
of 1997.
 
     The volume of oil sold decreased approximately 14% over the comparable
period last year, and the average price per barrel of oil decreased to $12.29
per barrel from $13.61 a year ago.
 
     Natural gas volumes sold decreased approximately 42% in the current period
from the comparable period last year. Natural gas volumes amounted to 175,595
Mcf this period and were 301,518 Mcf for the comparable period last year. The
average price increased $.71 to $3.35 per Mcf, from $2.64 a year ago.
 
     Income from the equity interest in Tidelands increased approximately 63% in
the current period over the comparable period last year.
 
     The quantities of oil and gas sold and the average prices for oil and gas
(including the equity in Tidelands) for the three month periods ended March 31,
1998 and 1997 are presented in the following table:
 
<TABLE>
<CAPTION>
                                                                1998       1997
                                                              --------    -------
<S>                                                           <C>         <C>
OIL
  Barrels sold..............................................    10,301     12,046
  Average price.............................................  $  12.29    $ 13.61
 
NATURAL GAS
  Mcf sold..................................................   175,595    301,518
  Average price.............................................  $   3.35    $  2.64
</TABLE>
 
     In addition to the quantities of oil and gas sold and the average prices
received for such oil and gas, two primary items affected income during the most
recent quarter. First, Marine Petroleum Corporation reduced by $52,474 the
previously-established account payable for possible royalty overpayments. This
amount increased income during the period. Secondly, Chevron requested that
Marine Petroleum Corporation repay royalties that Chevron believes it
incorrectly paid to Marine Petroleum Corporation during 1995, 1996 and 1997.
Although this matter has not yet been resolved, Marine Petroleum Corporation
increased the account payable for possible royalty overpayments by $167,000.
This increase in the amount of the account payable and reduced income by an
equivalent amount.
 
                                        7
<PAGE>   8
 
RESULTS OF OPERATIONS -- NINE MONTHS ENDED MARCH 31, 1998
 
     Net income for the nine months ended March 31, 1998 amounted to $2,338,573,
which was approximately 23% less than the $3,019,146 realized in the comparable
period of 1997.
 
     The volume of oil sold decreased approximately 14% over the comparable
period last year, and the average price per barrel of oil (based on actual
receipts) decreased to $16.92 per barrel from $24.17 a year ago.
 
     Natural gas volumes sold decreased approximately 21% in the current period
from the comparable period last year. Natural gas volumes amounted to 600,580
Mcf this period and were 758,543 Mcf for the comparable period last year. The
average price increased $.23 to $2.68 per Mcf, from $2.45 a year ago.
 
     Income from the equity interest in Tidelands increased approximately 65% in
the current period over the comparable period last year. See " -- General."
 
     The quantities of oil and gas sold and the average prices for oil and gas
(including the equity in Tidelands) for the six month periods ended December 31,
1997 and 1996 are presented in the following table:
 
<TABLE>
<CAPTION>
                                                                1998        1997
                                                              --------    --------
<S>                                                           <C>         <C>
OIL
  Barrels sold..............................................    47,166      54,732
  Average price.............................................  $  16.92    $  21.84
 
NATURAL GAS
  Mcf sold..................................................   600,586     758,543
  Average price.............................................  $   2.68    $   2.45
</TABLE>
 
     In addition to the quantities of oil and gas sold and the average prices
received for such oil and gas, two primary items affected income during the nine
months ended March 31, 1998. First, Marine Petroleum Corporation reduced by
$52,474 the previously-established account payable for possible royalty
overpayments. This amount increased income during the period. Secondly, Chevron
requested that Marine Petroleum Corporation repay royalties that Chevron
believes it incorrectly paid to Marine Petroleum Corporation during 1995, 1996
and 1997. Although this matter has not yet been resolved, Marine Petroleum
Corporation increased the account payable for possible royalty overpayments by
$167,000. This increase in the amount of the account payable and reduced income
by an equivalent amount.
 
PART II. OTHER INFORMATION
 
     There were no events reportable under Part II of the Quarterly Report on
Form 10-Q.
 
                                        8
<PAGE>   9
 
                                   SIGNATURES
 
     Pursuant to the requirements of the Securities Exchange Act of 1934, the
registrant has duly caused this report to be signed on its behalf by the
undersigned, thereunto duly authorized.
 
                                            MARINE PETROLEUM TRUST
 
                                            NationsBank of Texas, N.A., Trustee
 
May 12, 1998                                By:      /s/ JANE J. SHEA
                                              ----------------------------------
                                                         Jane J. Shea
                                                        Vice President
 
May 12, 1998                                         /s/ R. RAY BELL
                                            ------------------------------------
                                                Principal Accounting Officer
 
                                        9
<PAGE>   10
                               INDEX TO EXHIBITS


<TABLE>
<CAPTION>
EXHIBIT
  NO.                   ITEM
-------                 ----
  <S>          <C>
  27           Financial Data Schedule
</TABLE>