Poziotinib in EGFR Exon 20 Mutant Advanced NSCLC

Inclusion Criteria: * Histologically or cytologically confirmed stage IV or recurrent solid tumor not amenable to curative intent therapy * Cohort 1 specific inclusion criteria: NSCLC with documented EGFR exon 20 mutation by one of the following Clinical Laboratory Improvement Act (CLIA) certified tests: OncoMine Comprehensive Assay (OCA), Guardant360 Assay (using plasma), or FoundationOne Assay or by a Food and Drug Administration (FDA) approved device using cobas EGFR mutation test version (v)2 or therascreen EGFR RGQ PCt kit; Mutations include D770\_N771insSVD, D770\_N771insNPG, V769\_D770insASV, H773\_V774insNPH, or any other exon 20 in-frame insertion or point mutation excluding T790M * Cohort 2 specific inclusion criteria: NSCLC with documented HER2 exon 20 mutation by a CLIA certified tests: Oncomine Comprehensive Assay (OCA), Guardant360 Assay (using plasma), or FoundationOne Assay; eligible mutations include A775\_G776insYVMA, G776\_V777insVC, or P780\_Y781insGSP, or any other in-frame exon 20 insertion mutation or point mutation including, but not limited to, L755S, G776V, and V777L * Cohort 3 specific inclusion criteria: NSCLC with documented EGFR exon 20 mutation (excluding T790M) by one of the following CLIA certified tests: OncoMine Comprehensive Assay (OCA), Guardant360 Assay (using plasma), or FoundationOne Assay or by an FDA approved device using cobas EGFR mutation test v2 or therascreen EGFR RGQ PCt kit; mutations include D770\_N771insSVD, D770\_N771insNPG, V769\_D770insASV, H773\_V774insNPH, or any other exon 20 in-frame insertion or point mutation excluding T790M * Cohort 4 specific inclusion criteria: solid tumor (excluding NSCLC) harboring EGFR or HER2 exon 20 mutations documented by one of the following CLIA certified tests: Oncomine Comprehensive Assay (OCA), Guardant360 Assay (using plasma), or FoundationOne Assay * Cohort 5 specific inclusion criteria: solid tumor harboring HER2 exon 19 mutation documented by one of the following CLIA certified tests: Oncomine Comprehensive Assay (OCA), Guardant360 Assay (using plasma), or FoundationOne Assay * Patients must have exhausted all FDA-approved standard of care options for locally advanced or metastatic disease, or are ineligible for FDA-approved standard of care options, or refusing to receive FDA-approved standard of care options. Previously untreated patients are allowed only in cohort 1 and 3 and are eligible only if EGFR Exon 20 mutation is confirmed using an FDA approved device/test such as cobas EGFR Mutation Test v2, therascreen EGFR RGQ PCR Kit or other FDA tests prior to study enrollment * Patient has adequate tumor tissue obtained from a biopsy or surgical procedure to enable molecular profiling for retrospective central laboratory confirmation of the mutation. If tissue is not available, the patient must have biopsy accessible disease and must be willing to undergo a biopsy prior to the study * Measurable disease by RECIST 1.1 * Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 * Ability to take pills by mouth * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 100,000/mcL * Hemoglobin \>= 9.0 g/dL * Total bilirubin =\< 2 x upper limit of normal (ULN) * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 institutional upper limit of normal of =\< 5 x ULN if liver metastases are present * Alkaline phosphatase =\< 2.5 institutional upper limit of normal of =\< 5 x ULN if liver metastases are present * Creatinine clearance \>= 50 mL/min/1.73 m\^2 by Cockcroft-Gault equation * Brain metastases are allowed, as long as they are stable and do not require treatment with anticonvulsants or escalating doses of steroids * Females of childbearing potential must have a negative serum or urine pregnancy test and must agree to use adequate contraception for the duration of the study and six months after; adequate contraception methods include: birth control pills (e.g. combined oral contraceptive pill), barrier protection (e.g. condom plus spermicide, cervical/vault cap or intrauterine device), and abstinence; females of childbearing potential are defined as those who are not surgically sterile (i.e., bilateral tubal ligation, bilateral oophorectomy, or complete hysterectomy) or postmenopausal (defined as 12 months with no menses without an alternative medical cause); women will be considered post-menopausal if they have been amenorrheic for the past 12 months without an alternative medical cause * The following age-specific requirements must also apply: * Women \< 50 years old: they would be considered post-menopausal if they have been amenorrheic for the past 12 months or more following cessation of exogenous hormonal treatments; the levels of luteinizing hormone (LH) and follicle-stimulating hormone (FSH) must also be in the post-menopausal range (as per the institution) * Women \>= 50 years old: they would be considered post-menopausal if they have been amenorrheic for the past 12 months or more following cessation of all exogenous hormonal treatments, or have had radiation-induced oophorectomy with the last menses \> 1 year ago, or have had chemotherapy-induced menopause with \> 1 year interval since last menses, or have had surgical sterilization by either bilateral oophorectomy or hysterectomy * Non-sterilized males who are sexually active with a female partner of childbearing potential must use adequate contraception for the duration of the study and 90 days after the last dose of study medication; adequate contraception methods include: birth control pills (e.g. combined oral contraceptive pill), barrier protection (e.g. condom plus spermicide, cervical/vault cap or intrauterine device), and abstinence * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * EGFR T790M mutation or any other acquired EGFR exon 20 mutation; patients with coexisting primary EGFR exon 20 and T790M mutations are eligible * Have received or are receiving an investigational medicinal product (IMP) or other systemic anticancer treatment within 2 weeks prior to the first dose of study treatment * Any concurrent chemotherapy, immunotherapy, biologic, or hormonal therapy for cancer treatment * Have known or suspected brain metastases or spinal cord compression, unless the condition has been asymptomatic, has been treated with surgery and/or radiation, and has been stable without requiring escalating corticosteroids nor anti-convulsant medications for at least 4 weeks prior to the first dose of study medication * Known hypersensitivity to poziotinib or history of allergic reactions attributed to compounds of similar chemical or biologic composition to poziotinib * Cardiac conditions as follows: * 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 cardiac ejection fraction \< 50% by either echocardiogram or multi-gated acquisition (MUGA) scan * Have any unresolved chronic toxicity with Common Terminology Criteria for Adverse Events (CTCAE) version 4.03 grade \>= 2, from previous anticancer therapy, except for alopecia * Patient is unable to take drugs orally due to disorders or diseases that may affect gastrointestinal function, such as inflammatory bowel diseases (e.g., Crohn's disease, ulcerative colitis) or malabsorption syndrome, or procedures that may affect gastrointestinal function, such as gastrectomy, enterectomy, or colectomy * Have any condition or illness that, in the opinion of the investigator, might compromise patient safety or interfere with the evaluation of the safety of the drug * Pregnant or breastfeeding women * History of another primary malignancy within 2 years prior to starting study treatment, except for adequately treated basal or squamous cell carcinoma of the skin or cancer of the cervix in situ * Recent major surgery within 4 weeks prior to starting study treatment, with the exception of surgical placement for vascular access * Male or female patients of reproductive potential who are not employing an adequate method of birth control; adequate contraception methods include: birth control pills (e.g. combined oral contraceptive pill), barrier protection (e.g. condom plus spermicide cervical/vault cap or intrauterine device), and abstinence * Uncontrolled intercurrent illness including, but not limited to, uncompensated respiratory, cardiac, hepatic, or renal disease, active infection (including hepatitis B, hepatitis C, human immunodeficiency virus \[HIV\], and active clinical tuberculosis), or renal transplant; ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, active peptic ulcer disease or gastritis, or psychiatric illness/social situations that would limit compliance with study requirements * Active bleeding disorders.