Birinapant for Advanced Ovarian, Fallopian Tube, and Peritoneal Cancer

* INCLUSION CRITERIA: Study participants will be eligible for study participation if they are/have: * Females greater than or equal to 18 years of age. Because no dosing or adverse event data are currently available on the use of birinapant in patients \< 18 years of age, children are excluded from this study, but will be eligible for future pediatric trials. * Able to understand and voluntarily sign a written informed consent, and are willing and able to comply with the protocol requirements including the requirement for biopsies for research purposes. * Advanced metastatic or unresectable epithelial ovarian cancer, primary peritoneal cancer or fallopian tube cancer that is relapsed and resistant or refractory to prior platinum-based standard care systemic regimen. Patients who are unable to receive further platinum, due to either allergic reaction or other medical reason, are eligible for the protocol. There is no limitation on the amount of prior therapies allowed. * Patients must be at least 4 weeks from previous therapy (chemotherapy, hormonal therapy, and radiation therapy, immunotherapy and monoclonal antibodies, alternative therapy or investigational therapeutic agents). Patients who have had cranial radiation therapy need to have completed it greater than or equal to 8 weeks prior to commencing on study. Patients are permitted to receive investigational imaging agents while on study. * Patients who have had major surgery must be fully recovered and require a recovery period of greater than or equal to 4 weeks prior to enrolling on study. * Histopathologic diagnosis must be confirmed in the Laboratory of Pathology (LP), National Cancer Institute (NCI). A block of the primary tumor, or access to recut slides is preferred. If this is unavailable, a recent resection specimen of a metastatic site is required for entry. In addition, access to archival formalin fixed paraffin embedded (FFPE) tumor blocks will be requested to perform correlative studies. Study participants with the following histologic epithelial ovarian cancer cell types are eligible: Serous adenocarcinoma, endometrioid adenocarcinoma, mucinous adenocarcinoma, undifferentiated carcinoma, clear cell adenocarcinoma, mixed epithelial carcinoma, transitional cell carcinoma, malignant Brenner's Tumor, or adenocarcinoma not otherwise specified (N.O.S). * Patients must have measurable disease by Response Evaluation Criteria in Solid Tumors defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as greater than or equal to 20 mm with conventional techniques or as greater than or equal to 10 mm with spiral computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam and a sentinel lesion adequate for core biopsy through percutaneous biopsy. See Section 13 for the evaluation of measurable disease. * Life expectancy greater than 3 months. * Eastern Cooperative Oncology Group (ECOG) performance status less than or equal to less than or equal to 2. * Adequate renal function, defined as serum creatinine less than or equal to 1.5 X upper limit of normal (ULN), or measured creatinine clearance greater than or equal to 60 ml/min/1.73m\^2. * Adequate hepatic function, defined as aspartate aminotransferase (AST) and alanine aminotransferase (ALT) levels less than or equal to 3 X ULN and total bilirubin \< 1.5 X ULN, unless known diagnosis of Gilberts syndrome. * Adequate bone marrow function, defined as absolute neutrophil (ANC) greater than or equal to 1,500/mm\^3 (greater than or equal to1.5 X10\^6/L), platelet count greater than or equal to 75,000/mm\^3 (greater than or equal to 75 X10\^6/L), and hemoglobin greater than or equal to 10 mg/dL (transfusion to obtain hemoglobin greater than or equal to 10 mg/dL within 24 hours prior to dosing is allowed). * Contraception is not a consideration as these patients have all had surgical removal of their reproductive organs. Pregnant women are excluded from this study because birinapant may have potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with birinapant, breastfeeding should be discontinued prior to enrollment. EXCLUSION CRITERIA: 1. Known or suspected diagnosis of human immunodeficiency virus or chronic active hepatitis B or C. Viral testing is not required. The reason for exclusion is insufficient evidence demonstrating safety of administration of birinapant in patients with human immunodeficiency virus (HIV), hepatitis B or C due to theoretical risk of unmasking or exacerbating these serious viral illnesses since birinapant may impair immunological function. Data are not currently available on risk of interaction with antiretroviral drugs. HIV-positive patients on combination antiretroviral therapy are ineligible because of the potential for pharmacokinetic interactions with birinapant. The potential immune suppressive effects and T-cell depletion associated with birinapant pose an additional increased risk to these patients. Appropriate studies will be undertaken in patients receiving combination antiretroviral therapy when indicated. 2. Symptomatic or uncontrolled brain metastases requiring current treatment (\< 8 weeks from last cranial radiation or \< 4 weeks from last steroids). (Patients with abnormal clinical exam or history will require a head CT or MRI to rule out or confirm brain metastases). 3. Impaired cardiac function or clinically significant cardiac disease including the following: 1. New York Heart Association grade III or IV congestive heart failure. 2. Myocardial infarction within the last 12 months prior to dosing with birinapant. 3. Subjects known to have impaired left ventricular ejection fraction (LVEF) according to institutional standards must be excluded. 4. Lack of recovery of prior adverse events to Grade less than or equal to1 severity (National Cancer Institute Common Terminology Criteria for Adverse Events \[NCI CTCAE\] v 4.03) (except alopecia) due to therapy administered prior to the initiation of study drug dosing. Stable persistent grade 2 peripheral neuropathy may be allowed as determined on a case-by-case basis at the discretion of the Investigator as birinapant has not been shown to cause or exacerbate peripheral neuropathy. 5. Known allergy to any of the formulation components of birinapant including citric acid monohydrate, sodium citrate dehydrate, and sodium chloride. 6. Any concurrent disease and/or medical condition that in the opinion of the Investigator would prevent the subjects participation, render the subject at excessive risk or limit the subjects compliance with the protocols required evaluations. 7. Patients with active infection will not be eligible, but may become eligible once infection has resolved and they are at least 7 days from completion of antibiotics. 8. Another previous or current malignancy within the last 5 years, with the exception of non-melanoma skin cancer, cervical carcinoma in situ curatively treated, ductal or lobular carcinoma in situ curatively treated and without ongoing therapeutic intervention. Patients with breast cancer (BRCA) 1 or 2 mutation, who have had a previous diagnosis of breast cancer are eligible if the breast cancer was diagnosed 5 years previously and distant or local recurrence of breast cancer has been outruled. 9. Concomitant chronic (daily or almost daily for greater than or equal to1 month prior) use of steroids or non-steroidal anti-inflammatory drugs (NSAIDS). Intermittent use of steroids as pre-medications is allowed. Based on research to date the tumor and tumor microenvironment production of tumor necrosis factor (TNF) alpha could promote anti-tumor activity. Theoretically, anti-inflammatories could blunt local production, limiting this possibly positive cofactor. 10. No concomitant use of complementary or alternative medication or other agents (investigational therapeutic agents) will be allowed without approval of a principal investigator (PI) or associate investigator (AI). Every effort will be made to maximize patient safety and minimize changes in chronic medications. 11. Patients with a recent history (within last 5 years) of autoimmune disease or inflammatory diseases will be excluded, for example, active rheumatoid arthritis,active inflammatory bowel disease or any chronic inflammatory conditions because birinapant synergizes with TNF in vitro. INCLUSION OF MINORITIES: Women from all racial/ethnic groups are eligible for this study if they meet the eligibility criteria. To date, there is no information that suggests that differences in drug metabolism or disease response would be expected in one group compared to another. Efforts will be made to extend accrual to a representative population, but in this preliminary study, a balance must be struck between patient safety considerations and limitations on the number of individuals exposed to potentially toxic and/or ineffective treatments on the one hand and the need to explore ethnic aspects of clinical research on the other hand. If differences in outcome that correlate to ethnic identity are noted, accrual may be expanded or a follow-up study may be written to investigate those differences more fully. This study will be recruited through internal referral, our local physician referral base, and through Cancer Hotline information.