Inclusion Criteria:  -Participants must have histologic or cytologic diagnosis of non-Hodgkin lymphoma, with the exclusion of small lymphocytic lymphoma/chronic lymphocytic leukemia.  Patients with mantle cell lymphoma are not eligible for the dose escalation part of the study. Inclusion of patients with mantle cell lymphoma to the dose expansion part of the study will be done after an amendment delineates a MCL - specific venetoclax ramp up and tumor lysis syndrome prophylaxis and monitoring regimen.  -Participants must have received ≥2 prior systemic therapies for their lymphoma.  -Participants must have measurable disease as defined by the 2014 Lugano Classification.  -Participants must meet clinical indications for treatment.  -ECOG performance status ≤ 2 (see Appendix I)  -Adequate bone marrow function, defined by the following laboratory parameters  Absolute neutrophil count of 1.0 x 109/L  Platelet count of 75 x 109/L; platelet count of 50 - 75 x 109/L are permitted in participants with marrow involvement by the lymphoma. Platelets must not have received a platelet transfusion in 7 days.  -Adequate organ function, defined by the following laboratory parameters  Adequate hepatic function, with transaminases (alanine aminotransferase [ALT], aspartate aminotransferase [AST], and gamma glutammyltransferase [GGT]) ≤ 2.5 times the upper limit of normal; Bilirubin ≤1.5 x ULN (unless bilirubin rise is due to Gilbert's syndrome or of non-hepatic origin) Serum creatinine ≤ 1.5 times the upper limit of normal. -For women of childbearing potential: agreement to remain abstinent (refrain from heterosexual intercourse) or use a contraceptive method with a failure rate of < 1% per year during the treatment period and for at least 30 days after the last dose of venetoclax and at least 9 months after the last dose of loncastuximab tesirine for women. A woman is considered to be of childbearing potential if she is postmenarcheal, has not reached a postmenopausal state (< 12 continuous months of amenorrhea with no identified cause other than menopause), and has not undergone surgical sterilization (removal of ovaries and/or uterus).  Examples of contraceptive methods with a failure rate of < 1% per year include bilateral tubal ligation, male sterilization, hormonal contraceptives that inhibit ovulation, hormone-releasing intrauterine devices, and copper intrauterine devices.  The reliability of sexual abstinence should be evaluated in relation to the duration of the clinical trial and the preferred and usual lifestyle of the patient. Periodic abstinence (e.g., calendar, ovulation, symptothermal, or postovulation methods) and withdrawal are not acceptable methods of contraception.  -For men: agreement to remain abstinent (refrain from heterosexual intercourse) or use contraceptive measures, and agreement to refrain from donating sperm, as defined below:  --With female partners of childbearing potential, men must remain abstinent or use a condom plus an additional contraceptive method that together result in a failure rate of < 1% per year during the treatment period and for at least 6 months after the last dose of loncastuximab. Men must refrain from donating sperm during this same period.  With pregnant female partners, men must remain abstinent or use a condom during the treatment period and for at least 6 months after the last dose of loncastuximab to avoid exposing the embryo.  The reliability of sexual abstinence should be evaluated in relation to the duration of the clinical trial and the preferred and usual lifestyle of the patient. Periodic abstinence (e.g., calendar, ovulation, symptothermal, or postovulation methods) and withdrawal are not acceptable methods of contraception.  Exclusion Criteria:  Prior treatment toxicities not resolved to grade <2 according to NCI CTCAE 5.0 (with the exception of alopecia or grade 2 sensory peripheral neuropathy). Patients with spontaneous tumor lysis syndrome. Autologous stem cell transplant within 30 days of start of study drug (C1D1). Allogeneic stem cell transplant within 60 days of start of study drug (C1D1). Women who are pregnant or breastfeeding. Active graft versus host disease Active autoimmune disease Known seropositive and requiring anti-viral therapy for human immunodeficiency (HIV) virus. Note: Testing is not mandatory to be eligible. Malabsorption syndrome or other condition that precludes enteral route of administration. Known allergy to both xanthine oxidase inhibitors and rasburicase. Allergy to only one of these agents does not constitute an exclusion criterion.  Use of strong CYP3A inhibitors or inducers.  --All medications that fall in these categories should be discontinued 7 days prior to the first dose of study drug.  Administration or consumption of any of the following within 3 days prior to the first dose of study drug:  Grapefruit or grapefruit products Seville oranges (including marmalade containing Seville oranges) Star fruit  Evidence of other clinically significant uncontrolled condition(s) including, but not limited to:  Uncontrolled and/or active systemic infection (viral, bacterial or fungal) Chronic hepatitis B virus (HBV) or hepatitis C (HCV) requiring treatment. Note: subjects with serologic evidence of prior vaccination to HBV (i.e. hepatitis B surface (HBs) antigen negative-, anti-HBs antibody positive and anti-hepatitis B core (HBc) antibody negative) or positive anti-HBc antibody from intravenous immunoglobulins (IVIG) may participate. Other uncontrolled conditions including uncontrolled cardiovascular disease or arrythmia, decompensated diabetes or COPD.