A Study of Poziotinib in Patients With Epidermal Growth Factor Receptor (EGFR) or Human Epidermal Growth Factor Receptor 2 (HER2) Activating Mutations in Advanced Malignancies

Key Inclusion Criteria: 1. Patients must be at least 18 years old. 2. Patients must have histologic or cytologic evidence of a malignant solid cancer that is either advanced or metastatic there must be no available therapy known to confer a reasonable likelihood of clinical benefit. 3. Patients with BC must have a HER2 activating mutation determined by next-generation sequencing (NGS) performed on either tumor or plasma samples and: * Immunohistochemistry (IHC) HER2-positive BC that has progressed on trastuzumab, pertuzumab, and trastuzumab emtansine (T-DM1) in the metastatic setting, unless there is recurrent disease within 12 months of adjuvant or neoadjuvant treatment. * IHC HER2-negative, estrogen receptor/progesterone receptor (ER/PR)-positive BC that has progressed on or after appropriate first-line endocrine therapy in the metastatic setting. * IHC HER2-negative, ER/PR-negative BC that has progressed after first-line treatment (any standard chemotherapy-based regimen) in the metastatic setting. 4. Patients with microsatellite instability-high (MSI-H) CRC must have had appropriate checkpoint inhibitor-based therapy. 5. Patient's tumor must be positive for an EGFR or HER2 mutation based on DNA testing of either tumor tissue or plasma samples. Patients with documented EGFR or HER2 mutations may be identified by local testing from participating sites using next generation sequencing tests. Patient has a solid tumor with at least one of the listed activating mutations: * Cohorts 1-3: HER2 Activating Mutations (at least one of the following) Furin-Like/Extracellular. S310F/Y Transmembrane. I655V, V659E, R678Q, V697L Kinase Domain. Exon 20 insertion, T733I, L755X, I767M, D769X, V773M, V777X, L786V, V842I, T862I, L869R. * Cohorts 4-5: EGFR Activating Mutations (at least one of the following) Extracellular \& Transmembrane: EGFRvIII, R108K, R222C, A289T, P596L, G598V Kinase Domain: Exon 20 insertion, E709X, E709\_T710del insD, L718X, G719X, I740\_K745dupIPVAIK, I740\_K745dup, V742I, L747X, E746\_A750del, A750P, S768I, S768I/V769L, S768I/V774M, L833V, V769M, V774M, R831C, R831H, L858R, L861Q, A864V. 6. Patients must have measurable disease. 7. Patients with central nervous system (CNS) metastases must have stable CNS disease and no evidence of growth on imaging for at least 4 weeks following radiation or other locoregional ablative therapy. CNS symptoms must be stable with no requirement for anti-seizure medications and/or \> 2 mg/day dexamethasone equivalent, except for patients with GBM (Cohort 4), in whom anti-seizure medications and/or up to 4 mg/day dexamethasone equivalent is allowed. 8. Patients must have an Eastern Cooperative Oncology Group (ECOG) performance status ≤ -1. 9. Patients must have adequate hematopoietic, renal, and liver functions. Exclusion Criteria: 1. Patient has an EGFR T790 mutation. 2. Patient has breast or gastric cancer without eligible HER2 mutations (see Inclusion Criteria). 3. Patients with GBM enrolled in Cohort 4 has been treated with an inhibitor of vascular endothelial growth factor (VEGF) inhibitor therapy (e.g., bevacizumab). 4. Patient requires treatment with a medication that is a strong inhibitor or inducer of CYP3A4 or CYP2D6 or has been treated with such medications within 15 days of poziotinib treatment. 5. Patient has ≥ Grade 2 skin disorders (rash), mucositis, or stomatitis within 15 days of poziotinib treatment. 6. Patient has a gastrointenstinal disorder or malabsorption that precludes oral drug treatment. 7. Patient has active liver or biliary tract disease (except for Gilbert's syndrome, asymptomatic biliary stones, liver metastasis, or stable chronic liver diseases). 8. Patient has a history of drug-induced pancreatitis. 9. Patient has a history of interstitial lung disease or pneumonitis. 10. Patient has a history of congestive heart failure Class III/IV according to the New York Heart Association Functional Classification or a serious cardiac arrhythmia requiring treatment. 11. Patient has a high risk of cardiac disease as determined by the Investigator. If patient is deemed to have a high cardiac risk, enrollment may be considered if an echocardiogram (ECHO) or multi-gated acquisition (MUGA) during Screening demonstrates a cardiac ejection fraction \>= 50%. 12. Patient has a QTc interval \> 470 ms. 13. Patient has a history of another malignancy within the 1 year prior to poziotinib treatment. Patients with the following concomitant neoplastic diagnoses are eligible: non-melanoma skin cancer, carcinoma in situ (including transitional cell carcinoma, cervical intraepithelial neoplasia, and melanoma in situ), organ-confined prostate cancer with no evidence of progressive disease.