Key Inclusion Criteria:

1. Participant must be willing and capable of giving written Informed Consent, adhering to dosing and visit schedules, and meeting all study requirements
2. Participant has histologically or cytologically confirmed locally advanced or metastatic non-small cell lung cancer (NSCLC) that is not amenable to treatment with curative intent
3. Prior treatment status:

1. Cohorts 1 and 2: Participant has had at least one prior systemic treatment for locally advanced or metastatic NSCLC
2. Cohorts 3 and 4: Participant is treatment-naïve for locally advanced or metastatic NSCLC and eligible to receive first-line treatment with poziotinib as determined by the Investigator. Adjuvant/neo-adjuvant therapies (chemotherapy, radiotherapy, or investigational agents) are permissible as long as they end at least 15 days prior to study entry.
3. Cohort 5: Participants who meet the criteria for enrollment in Cohorts 1 to 4, but the enrollment in the respective cohort has been closed
4. Cohort 6: Participant with EGFR mutation-positive NSCLC who progressed while on treatment with first-line osimertinib
5. Cohort 7: Participant has had at least one prior systemic treatment for locally advanced or metastatic NSCLC
6. Specific mutations:

1. Cohort 1 and 3: Documented EGFR exon 20 insertion mutation
2. Cohort 2 and 4: Documented HER2 exon 20 insertion mutation
3. Cohort 5: Documented EGFR or HER2 exon 20 insertion mutations
4. Cohort 6: Documented acquired EGFR mutation (tested after osimertinib progression)
5. Cohort 7: Documented EGFR or HER2 activating mutations
6. Participant has adequate organ function at Baseline

Key Exclusion Criteria:

1. Participant has had previous treatment with poziotinib or any other EGFR or HER2 exon 20 insertion mutation-selective tyrosine kinase inhibitor (TKI) prior to study participation. The currently approved TKIs (ie, erlotinib, gefitinib, afatinib, osimertinib) are not considered to be exon 20 insertion-selective and are permissible (Cohorts 1 and 2).
2. Participant is concurrently receiving chemotherapy, biologics, immunotherapy for cancer treatment; systemic anti-cancer treatment or investigational treatment should not be used within 2 weeks or 5 half lives, whichever is longer; local radiation therapy for bone pain may be allowed
3. Participant has had other malignancies within the past 3 years, except for stable non-melanoma skin cancer, fully-treated and stable, early-stage prostate cancer, or carcinoma in situ of the cervix or breast without need of treatment
4. Participant is pregnant or breast-feeding