Inclusion Criteria:  Patient, or patient's authorized representative, must be willing and capable of giving written informed consent and must be able to adhere to dosing and visit schedules as well as meet all study requirements  Patient has histologically or cytologically confirmed non-small cell lung cancer (NSCLC) not amenable to curative intent therapy or stage IV NSCLC Documented epidermal growth factor receptor (EGFR) exon 20 point or insertion mutation using a Food and Drug Administration (FDA)-approved in vitro diagnostic test (ie, cobas EGFR mutation test version [v] 2 or therascreen EGFR RGQ polymerase chain reaction [PCR] kit), a Clinical Laboratory Improvement Act (CLIA) certified test (eg, OncoMine Comprehensive Assay (OCA), Guardant360 Assay [using plasma], or FoundationOne Assay), or similarly accredited test for tissue or plasma Brain metastases are allowed, as long as they are stable and do not require treatment with anticonvulsants or escalating doses of steroids Previously untreated and/or any number of prior lines of therapy for metastatic disease are allowed in the dose expansion part. Previously untreated patients are now allowed in the dose finding portion of the study Patient is at least 18 years of age Patient has measurable disease, as per the Response Evaluation Criteria in Solid Tumors (RECIST, version 1.1) Patient has an Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 Patient has recovered from prior systemic therapy for metastatic disease to grade =< 1 for non-hematologic toxicities (except for grade =< 2 peripheral neuropathy) Leukocytes >= 3.0 x 10^9/L Absolute neutrophil count (ANC) must be >= 1.5 x 10^9/L Platelet count >= 100 x 10^9/L Hemoglobin >= 9.0 g/dL Total bilirubin =<1.5 x upper limit of normal (ULN); if hepatic metastases are present, =< 2.5 x ULN Aspartate aminotransferase/serum glutamic-oxaloacetic transaminase (AST/SGOT), alanine aminotransferase/serum glutamic-pyruvic transaminase (ALT/SGPT), and gamma-glutamyltransferase (GGT) =< 2.5 x ULN; if hepatic metastases are present, =< 0.5 x ULN Creatinine clearance >= 50 mL/min Adequate coagulation function as defined by international normalized ratio (INR) 1.5 and a partial thromboplastin time (PTT) (PTT/activated partial thromboplastin time [aPTT]) < 1.5 x ULN. Patients on full-dose anticoagulation must be on a stable dose (minimum duration 14 days) of oral anticoagulant or low molecular weight heparin (LMWH). If receiving warfarin, the patient must have an INR =< 3.0. For heparin and LMWH there should be no active bleeding (that is, no bleeding within 14 days prior to first dose of protocol therapy) or pathological condition present that carries a high risk of bleeding (for example, tumor involving major vessels or known varices) Patient is willing to practice 2 forms of contraception, one of which must be a barrier method, from study entry until at least 30 days after the last dose of study combination Females of childbearing potential must have a negative pregnancy test within 7 days prior to enrollment. Females who are postmenopausal for at least 1 year (defined as more than 12 months since last menses) or who are surgically sterilized do not require this test The patient's urinary protein is =< 1+ on dipstick or routine urinalysis (UA; if urine dipstick or routine analysis is >= 2+, a 24-hour urine collection for protein must demonstrate < 1000 mg of protein in 24 hours to allow participation in this protocol)  Exclusion Criteria:  • Patient has EGFR T790M mutation or other acquired EGFR exon 20point mutation following prior treatment with an EGFR-tyrosine kinase inhibitor (TKI)  Previous treatment with poziotinib or ramucirumab Patient is concurrently receiving chemotherapy, biologics, immunotherapy for cancer treatment; systemic anti-cancer treatment or investigational treatment should not be used within 2 weeks; local radiation therapy for bone pain may be allowed Patient has a history of congestive heart failure (CHF) class III/IV according to the New York Heart Association (NYHA) functional classification or serious cardiac arrhythmias requiring treatment Patient has a high risk of cardiac disease, as determined by the investigator, may undergo either echocardiogram (ECHO) or multi-gated acquisition (MUGA) during screening and if the patient has a cardiac ejection fraction < 50%, the patient will be excluded Patient has a history of other malignancies within the last 3 years, except for non-melanoma skin cancer or carcinoma in situ of the cervix Patient is confirmed to have clinically significant or recent (within 14 days prior to starting treatment) acute gastrointestinal disease presenting as diarrhea and/or coloenteritis as a main symptom (ie, acute enteritis, malabsorption, or Common Terminology Criteria for Adverse Events [CTCAE, version 5.0] grade 2 or above diarrhea due to other etiologies) Patient is unable to take drugs orally due to disorders or diseases that may affect gastrointestinal (GI) function, such as inflammatory bowel diseases (eg, Crohn's disease, ulcerative colitis) or malabsorption syndrome, or procedures that may affect gastrointestinal function, such as gastrectomy, enterectomy, or colectomy Patient has an active liver disease or biliary tract disease (except for Gilbert's disease, asymptomatic biliary stones, liver metastasis, or stabilized chronic liver diseases). Patients with liver cirrhosis at a level of Child-Pugh B (or worse) or history of hepatic encephalopathy or clinically meaningful ascites resulting from cirrhosis are excluded. Patient has known hypersensitivity to poziotinib or ramucirumab Patient has an active uncontrolled infection, underlying medical condition, or other serious illness that would impair the ability of the patient to receive protocol treatment Patient has experienced any grade 3-4 GI bleeding within 3 months prior to first dose of protocol therapy. Patient has a history of deep vein thrombosis (DVT), pulmonary embolism (PE), or any other significant thromboembolism (venous port or catheter thrombosis or superficial venous thrombosis are not considered "significant") during the 3 months prior to first dose of protocol therapy The patient has experienced any arterial thromboembolic events, including but not limited to myocardial infarction, transient ischemic attack, cerebrovascular accident, or unstable angina, within 6 months prior to first dose of protocol therapy The patient has uncontrolled or poorly-controlled hypertension (> 160 mmHg systolic or > 100 mmHg diastolic for > 4 weeks) despite standard medical management If they experience hemoptysis (defined as bright red blood or >= 1/2 teaspoon) within 2 months prior to first dose of protocol therapy or with radiographic evidence of intratumor cavitation or has radiologically documented evidence of major blood vessel invasion or encasement by cancer Patient has had recent major surgery within 28 days prior to starting study treatment, or minor surgery/subcutaneous venous access device placement within 7 days prior to the first dose of protocol therapy. The patient has elective or planned major surgery to be performed during the course of the clinical trial. The patient has a prior history of GI perforation/fistula (within 6 months of first dose of protocol therapy) or risk factors for perforation The patient has a serious or nonhealing wound, ulcer, or bone fracture within 28 days prior to first dose of protocol therapy Patient has EGFR T790M mutation or other acquired EGFR exon 20point mutation following prior treatment with an EGFR-tyrosine kinase inhibitor (TKI) Previous treatment with poziotinib or ramucirumab Patient is concurrently receiving chemotherapy, biologics, immunotherapy for cancer treatment; systemic anti-cancer treatment or investigational treatment should not be used within 2 weeks; local radiation therapy for bone pain may be allowed Patient has a history of congestive heart failure (CHF) class III/IV according to the New York Heart Association (NYHA) functional classification or serious cardiac arrhythmias requiring treatment Patient has a high risk of cardiac disease, as determined by the investigator, may undergo either echocardiogram (ECHO) or multi-gated acquisition (MUGA) during screening and if the patient has a cardiac ejection fraction < 50%, the patient will be excluded Patient has a history of other malignancies within the last 3 years, except for non-melanoma skin cancer or carcinoma in situ of the cervix Patient is confirmed to have clinically significant or recent (within 14 days prior to starting treatment) acute gastrointestinal disease presenting as diarrhea and/or coloenteritis as a main symptom (ie, acute enteritis, malabsorption, or Common Terminology Criteria for Adverse Events [CTCAE, version 5.0] grade 2 or above diarrhea due to other etiologies) Patient is unable to take drugs orally due to disorders or diseases that may affect gastrointestinal (GI) function, such as inflammatory bowel diseases (eg, Crohn's disease, ulcerative colitis) or malabsorption syndrome, or procedures that may affect gastrointestinal function, such as gastrectomy, enterectomy, or colectomy Patient has an active liver disease or biliary tract disease (except for Gilbert's disease, asymptomatic biliary stones, liver metastasis, or stabilized chronic liver diseases). Patients with liver cirrhosis at a level of Child-Pugh B (or worse) or history of hepatic encephalopathy or clinically meaningful ascites resulting from cirrhosis are excluded. Patient has known hypersensitivity to poziotinib or ramucirumab Patient has an active uncontrolled infection, underlying medical condition, or other serious illness that would impair the ability of the patient to receive protocol treatment Patient has experienced any grade 3-4 GI bleeding within 3 months prior to first dose of protocol therapy. Patient has a history of deep vein thrombosis (DVT), pulmonary embolism (PE), or any other significant thromboembolism (venous port or catheter thrombosis or superficial venous thrombosis are not considered "significant") during the 3 months prior to first dose of protocol therapy The patient has experienced any arterial thromboembolic events, including but not limited to myocardial infarction, transient ischemic attack, cerebrovascular accident, or unstable angina, within 6 months prior to first dose of protocol therapy The patient has uncontrolled or poorly-controlled hypertension (> 160 mmHg systolic or > 100 mmHg diastolic for > 4 weeks) despite standard medical management If they experience hemoptysis (defined as bright red blood or >= 1/2 teaspoon) within 2 months prior to first dose of protocol therapy or with radiographic evidence of intratumor cavitation or has radiologically documented evidence of major blood vessel invasion or encasement by cancer Patient has had recent major surgery within 28 days prior to starting study treatment, or minor surgery/subcutaneous venous access device placement within 7 days prior to the first dose of protocol therapy. The patient has elective or planned major surgery to be performed during the course of the clinical trial. The patient has a prior history of GI perforation/fistula (within 6 months of first dose of protocol therapy) or risk factors for perforation The patient has a serious or nonhealing wound, ulcer, or bone fracture within 28 days prior to first dose of protocol therapy Patient is pregnant or breast-feeding The patient is receiving chronic antiplatelet therapy, including dipyridamole or clopidogrel, or similar agents. Once-daily aspirin use (maximum dose 325 mg/day) is permitted The presence of interstitial lung disease, drug-related pneumonitis, or radiation pneumonitis at screening