Mirvetuximab Soravtansine and Rucaparib Camsylate in Treating Participants With Recurrent Endometrial, Ovarian, Fallopian Tube or Primary Peritoneal Cancer

Inclusion Criteria: * Patients must have histologically or cytologically confirmed: * Recurrent endometrial cancer (all histologies, including carcinosarcoma) * Recurrent ovarian, primary peritoneal (female only), or fallopian tube cancer * all histologies except low grade serous or clear cell carcinoma unless the patient has a known somatic or germline breast cancer (BRCA) mutation disease that is metastatic and for which standard curative measures do not exist or are no longer effective * For the dose escalation portion of the trial, patients with available therapies known to confer clinical benefit (platinum sensitive ovarian cancer) must be excluded * For the dose expansion cohort, patients with recurrent endometrial cancer, recurrent BRCA mutated ovarian cancer (except first-recurrence platinum sensitive ovarian cancer), and platinum resistant ovarian cancer are eligible * Patients must have confirmation of folate receptor-a (FR-alpha) positivity by immunohistochemistry (IHC) (? 25% of tumor staining at ? 2 + intensity) on archival tissue or recent biopsy. * Patients must be willing and able to undergo tissue biopsy for research * If tumor tissue obtained from the biopsy is deemed inadequate, and the patient is unwilling or unable to have another biopsy, the patient may be considered for enrollment if archival tumor tissue is provided and deemed of adequate quality; this must occur prior to any treatment with rucaparib or mirvetuximab soravtansine * If biopsy is deemed unsafe to attempt or to perform, and if archival tumor tissue is available and deemed of adequate quality, the patient may enroll on trial * Biopsy must be of solid tumor tissue; ascites is not acceptable * Patients must have measurable disease per Response Evaluation Criteria in Solid Tumors (RECIST) guideline (version 1.1) * Prior therapy: * Patients may have received unlimited prior treatment for the dose escalation part * For the expansion cohort patients must have ? 4 prior lines of chemotherapy * Hormonal therapy does not count towards total lines of therapy * Maintenance therapy is considered part of the preceding regimen if one or more of the same drugs are continued * Neoadjuvant and adjuvant chemotherapy are considered one regimen if they are a continuation of the same regimen with interval debulking surgery * Prior treatment with folate receptor (FR) targeting investigational agents is allowed for dose escalation provided that such treatment was not discontinued due to adverse events; prior FR-targeting investigational agents are not allowed for patients in the expansion cohort * Patients with recurrent endometrial, ovarian, fallopian tube or primary peritoneal cancer must have received at least one platinum-based chemotherapy regimen * Patients who have received prior taxanes, including weekly taxanes are allowed * Patients previously treated with a poly adenosine diphosphate (ADP) ribose polymerase (PARP) inhibitor may be enrolled provided: * PARP inhibitor was not the most recent treatment * PARP inhibitor treatment was discontinued \> 6 months before the first planned dose of rucaparib * Time from prior therapy: * Systemic anti-neoplastic therapy: five half-lives or four weeks, whichever is shorter (6 weeks for nitrosoureas or mitomycin C) * Hormonal therapy is not considered anti-neoplastic therapy * Radiotherapy: wide-field radiotherapy (e.g. \> 30% of marrow-bearing bones) completed at least four weeks, or focal radiation completed at least two weeks, prior to starting study treatment * Eastern cooperative oncology group (ECOG) performance status ? 1 and life expectancy \> 12 weeks * Leukocytes ? 2,000/mcL * Absolute neutrophil count ? 1,500/mcL * Platelets ? 100,000/mcL * Hemoglobin ? 9.0 g/dL * Total bilirubin ? 1.5 x upper limit of normal (ULN) (Patients with documented diagnosis of Gilbert syndrome are eligible if total bilirubin \< 3.0 x ULN) * Aspartate aminotransferase (AST) serum glutamic-oxaloacetic transaminase (SGOT)/ alanine aminotransferase (ALT) serum glutamate pyruvate transaminase (SGPT) ? 2.5 ? institutional upper limit of normal * Creatinine ? 1.5 x upper limit of normal (ULN) OR creatinine clearance ? 50 mL/min/1.73 m\^2 (using Cockroft Gault Formula) for patients with creatinine levels above institutional normal * Corrected QT (QTc) interval ? 470 msec on screening electrocardiogram (ECG) * Major surgery must have been completed ? 4 weeks prior to starting treatment day 1; patient must be sufficiently recovered and stable from surgery prior to treatment day 1 * Women of child-bearing potential must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately; patients must have a negative pregnancy test (urine and/or serum) prior to enrollment * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Patients with available therapies known to confer clinical benefit (platinum sensitive recurrent ovarian cancer) must be excluded from the dose escalation portion * For the dose expansion cohort patients with first-recurrence platinum-sensitive ovarian cancer must be excluded * Patients who have had chemotherapy or radiotherapy within 4 weeks or five half-lives whichever is shorter (6 weeks for nitrosoureas or mitomycin C) prior to entering the study * Patients who have not recovered from adverse events due to prior anti-cancer therapy (i.e., have residual drug related toxicities \> grade 1) except for alopecia and grade 2 fatigue * Patients who are receiving any other investigational agents * Primary platinum refractory disease (disease progression on first platinum treatment or recurrence within 3 months of completing first platinum regimen) * Patients with clear cell or low grade ovarian cancer unless the patient has a known germline or somatic breast cancer (BRCA) mutation or a mutation in another homologous recombination gene * Any known gastrointestinal disorder determined by the investigator that interferes with the absorption of rucaparib * Symptomatic or uncontrolled brain metastases requiring concurrent treatment, inclusive of but not limited to surgery, radiation and/or corticosteroids; patients with treated brain metastases are eligible as long as they completed prior brain radiation therapy more than 14 days prior to first dose of study therapy, are not experiencing seizures and are not receiving steroids for symptomatic brain metastases (for at least 7 days prior to first dose of study treatment) * History of leptomeningeal carcinomatosis * Subjects with a known history of uncontrolled seizures * History of allergic reactions attributed to compounds of similar chemical or biologic composition to rucaparib, mirvetuximab soravtansine or monoclonal antibodies * Uncontrolled inter-current illness including, but not limited to: * Ongoing or active infection (requiring IV antibiotics within 2 weeks of study enrollment) * Symptomatic/uncontrolled congestive heart failure (New York heart association \> class II) * Unstable angina pectoris * Recent myocardial infarction (\< 6 months) * Uncontrolled cardiac arrhythmia * Uncontrolled hypertension (? Common Terminology Criteria for Adverse Events \[CTCAE\] v4.03 grade 3) * Prior history of hypertensive crisis or hypertensive encephalopathy * Hemorrhagic or ischemic stroke \< 6 months * Clinically-significant vascular disease (e.g. aortic aneurysm, or dissecting aneurysm), severe aortic stenosis clinically significant peripheral vascular disease, or ? grade 3 cardiac toxicity following prior chemotherapy * Interstitial lung disease (ILD) * Active peptic ulcer disease or gastritis interfering with the absorption of rucaparib * Active bleeding diatheses including any subject known to have evidence of acute or chronic hepatitis B, hepatitis C * Active varicella zoster infection, cytomegalovirus infection, or history of tuberculosis * Psychiatric illness/social situations that would limit compliance with study requirements or compromise the ability of the subject to give written informed consent * Active or chronic corneal disorder, including but not limited to the following: Sjogren?s syndrome, Fuchs corneal dystrophy (requiring treatment), history of corneal transplantation, active herpetic keratitis, and also active ocular conditions requiring on-going treatment/monitoring such as wet age-related macular degeneration requiring intravitreal injections, active diabetic retinopathy with macular edema, presence of papilledema, acquired monocular vision, and any preexisting active conjunctival disease * History of neurological conditions that would confound assessment of treatment-emergent neuropathy * History of multiple sclerosis or other demyelinating disease and/or Eaton-Lambert syndrome (para-neoplastic syndrome) * Previous clinical diagnosis of non-infectious pneumonitis * History or evidence of thrombotic disorders within 6 months before first study treatment unless stable on anticoagulation for \> 3 months * Required used of folate-containing supplements (e.g. folate deficiency) * Has a known additional malignancy that is progressing or required active treatment within 3 years of first dose of study treatment; exceptions include basal cell carcinoma of the skin, squamous cell carcinoma of the skin that has undergone potentially curative therapy or other in situ cancers * Pregnant women are excluded from this study; breastfeeding should be discontinued if the mother is treated with rucaparib and mirvetuximab soravtansine * Women who are breastfeeding * Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible * Any condition that, in the opinion of the investigator, would interfere with evaluation of study treatment or interpretation of patient safety or study results