Inclusion Criteria:

          -  Histologically or cytologically confirmed stage IV or recurrent solid tumor not
             amenable to curative intent therapy

          -  Cohort 1 specific inclusion criteria: 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; eligible mutations include
             D770_N771insSVD, D770_N771insNPG, V769_D770insASV, H773_V774insNPH, or any other exon
             20 in-frame insertion or point mutation excluding T790M

          -  Patients must be previously treated with one or more regimens of systemic therapy for
             locally advanced or metastatic disease; previously untreated patients are eligible
             only if EGFR Exon 20 mutation is confirmed using an FDA approved device: cobas EGFR
             mutation test v2 or therascreen EGFR RGQ polymerase chain reaction (PCR) Kit prior to
             study enrollment

          -  Cohort 2 specific inclusion criteria: documented HER2 exon 20 mutation by a CLIA
             certified laboratory; 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

          -  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 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 effective
             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.