Pembrolizumab-based Therapy in Previously Treated High Grade Neuroendocrine Carcinomas

Inclusion Criteria: 1. Be willing and able to provide written informed consent for the trial. 2. Be at least 18 years of age on day of signing informed consent. 3. Have a histologically proven locally advanced or metastatic high grade (G3) poorly differentiated neuroendocrine carcinoma (NEC). 1. Includes small cell and large cell neuroendocrine carcinoma of unknown primary or any extrapulmonary site (and poorly differentiated NEC, not otherwise specified) 2. Includes neuroendocrine prostate cancer (de novo or treatment-emergent) of prostate if small cell or large cell histology (histologic evidence of both adenocarcinoma and neuroendocrine carcinoma may be present in same patient). 3. Other mixed tumors, e.g. mixed neuroendocrine neoplasms (MINENs) with NEC plus adenocarcinoma, squamous or acinar cell component are allowed if the high grade (small or large cell) NEC component comprises \>50% of the original sample or subsequent biopsy. 4. Have progressed during or after completion of first line systemic chemotherapy. 1. No limit to the number of prior chemotherapy regimens. 2. Early progression on/after adjuvant chemotherapy counts as firstline therapy. 5. Have at least one measurable disease based on RECIST 1.1. 6. Patients must agree to have a biopsy of primary tumor or metastatic tissue at baseline, and there must be a lesion that can be biopsied with acceptable clinical risk (as judged by the investigator). 1. Patients with unsuccessful baseline biopsies may undergo an additional biopsy attempt (at the same or a different site, determined by the investigator). 2. For patients with an intact primary and no metastatic site that can be safely biopsied, biopsy of the primary is acceptable, but must be approved by the principal investigator. 3. Baseline tumor biopsy may be omitted if the tumor is inaccessible and/or a biopsy is not thought to post exceptionally high procedural risk due to location or other factors 7. Have a performance status of 0 or 1 on the Eastern Cooperative Oncology Group (ECOG) Performance Scale. 8. Have a life expectancy of greater than 3 months. 9. Demonstrate adequate organ function, all screening labs should be performed within 14 days of treatment initiation. * Absolute neutrophil count (ANC) \>=1,500 /microliter (mcL) * Platelets \>=100,000 / mcL * Hemoglobin \>= 9 g/dL or \>=5.6 mmol/L without transfusion or erythropoietin (EPO) dependency (within 7 days of assessment) * Serum creatinine OR Measured or calculated creatinine clearance (CrCl) (Creatinine clearance should be calculated per institutional standard. Glomerular filtration rate (GFR) can also be used in place of creatinine or CrCl \<=1.5 X upper limit of normal (ULN) OR \>=60 mL/min for subject with creatinine levels \> 1.5 X institutional ULN * Serum total bilirubin \<= 1.5 X ULN OR Direct bilirubin \<= ULN for subjects with total bilirubin levels \> 1.5 ULN * aspartate aminotransferase (AST) / serum glutamic-oxaloacetic transaminase (SGOT) and alanine aminotransferase (ALT)/ serum glutamic-pyruvic transaminase (SGPT) \<=2.5 X ULN OR ≤ 5 X ULN for subjects with liver metastases * Albumin \>2.5 g/dL * International Normalized Ratio (INR) or Prothrombin Time (PT) Activated Partial Thromboplastin Time (aPTT) \<=1.5 X ULN unless subject is receiving anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants. 10. Female subject of childbearing potential should have a negative urine or serum pregnancy within 72 hours prior to receiving the first dose of study medication. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required. 11. Female subjects of childbearing potential should be willing to use 2 methods of birth control or be surgically sterile, or abstain from heterosexual activity for the course of the study through 120 days after the last dose of study medication. Subjects of childbearing potential are those who have not been surgically sterilized or have not been free from menses for \> 1 year. * Male subjects should agree to use an adequate method of contraception starting with the first dose of study therapy through 120 days after the last dose of study therapy. * Note: Abstinence is acceptable if this is the usual lifestyle and preferred contraception for the subject. Exclusion Criteria: 1. Has Merkel cell carcinoma, small cell lung carcinoma, or large cell NEC of lung * Intermediate grade neuroendocrine tumors are excluded * Well differentiated Grade 3 neuroendocrine tumors are excluded * Metastatic high-grade prostate carcinoma with evidence of focal neuroendocrine differentiation on prostate biopsy (e.g., positive chromogranin staining by immunohistochemistry) without small cell or large cell NEC morphology are excluded, as are neuroendocrine prostate cancers with phenotype intermediate between adenocarcinoma and small cell * Atypical and typical bronchial carcinoids and well differentiated G1 and G2 gastroenteropancreatic (GEP) neuroendocrine tumors (NET) (GEP NETs) are excluded. 2. Is currently participating and receiving study therapy or has participated in a study of an investigational agent and received study therapy or used an investigational device within 4 weeks of the first dose of treatment. 3. Has a diagnosis of immunodeficiency 4. Is receiving systemic steroid therapy or any other form of immunosuppressive therapy within 7 days prior to the first dose of trial treatment. \- Physiologic doses of steroids (e.g. =\< 10 mg prednisone/day or equivalent) are allowed 5. Has a known history of active Bacillus Tuberculosis (TB). 6. History of or high suspicion of Gilbert's disease (safety run-in, Part B only) 7. Hypersensitivity to pembrolizumab or any of its excipients. 8. Has had a prior anti-cancer monoclonal antibody (mAb) within 4 weeks prior to study Day 1 or who has not recovered (i.e., ≤ Grade 1 or at baseline) from adverse events due to agents administered more than 4 weeks earlier. 9. Documented progression on and/or intolerance/hypersensitivity to both paclitaxel and irinotecan (Part B only) 10. Has had prior chemotherapy, targeted small molecule therapy, or radiation therapy within 2 weeks prior to study Day 1 or who has not recovered (i.e., \<= Grade 1 or at baseline) from adverse events due to a previously administered agent. * Note: Subjects with ≤ Grade 2 neuropathy are an exception to this criterion and may qualify for the study. * Note: If subject received major surgery, they must have recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy. * Concurrent somatostatin analog therapy is allowed (for control of hormone excess) provided patient has been on stable dose for at least two months and tumor progression has been documented * Continuation of androgen deprivation therapy (ADT) allowed for patients with neuroendocrine prostate cancer (in the setting of castration-resistant prostate cancer, CRPC) 11. Has a known additional malignancy that is progressing or requires active treatment. Exceptions include basal cell carcinoma of the skin or squamous cell carcinoma of the skin that has undergone potentially curative therapy or in situ cervical cancer. 12. Has known active central nervous system (CNS) metastases and/or carcinomatous meningitis. Patients with asymptomatic suspected brain metastases (or small lesions of uncertain significance) \<1 cm that do not require focal therapy are eligible. (Follow up imaging will be allowed on study, and focal radiation with continuation of protocol therapy allowed if there is progressive disease in the brain and systemic imaging shows stable disease/response). \- Subjects with previously treated brain metastases may participate provided they are stable (without evidence of progression by imaging for at least four weeks and any neurologic symptoms have returned to baseline), they have no evidence of new or enlarging brain metastases (confirmed by imaging within 28 d of the first dose of trial treatment), and they are not using steroids for at least 7 days prior to trial treatment. This exception does not include carcinomatous meningitis, which is excluded regardless of clinical stability. 13. Has active autoimmune disease that has required systemic treatment in the past 2 years (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs). \- Replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment. 14. Has a history of (non-infectious) pneumonitis/ interstitial lung disease that required steroids or has current pneumonitis/interstitial lung disease. 15. Has an active infection requiring systemic therapy. 16. Has a history or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the trial, interfere with the subject's participation for the full duration of the trial, or is not in the best interest of the subject to participate, in the opinion of the treating investigator. 17. Has known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial. 18. Is pregnant or breastfeeding, or expecting to conceive or father children within the projected duration of the trial, starting with the pre-screening or screening visit through 120 days after the last dose of trial treatment. 19. Has received prior therapy with an anti-Programmed Death 1 (PD-1), anti-Programmed Death Ligand 1 (PD-L1), or anti-PD-L2 agent. 20. Has a known history of Human Immunodeficiency Virus (HIV) (HIV 1/2 antibodies). 21. Has known active Hepatitis B virus (e.g., HBsAg reactive) or Hepatitis C virus (HCV)(e.g., HCV RNA \[qualitative\] is detected). 22. Has received a live vaccine or live-attenuated vaccine within 30 days prior to the first dose of study drug. Administration of killed vaccines is allowed.