DEC-205/NY-ESO-1 Fusion Protein CDX-1401, Poly ICLC, and IDO1 Inhibitor INCB024360 in Treating Patients With Ovarian, Fallopian Tube, or Primary Peritoneal Cancer in Remission

Inclusion Criteria:

* Eligible patients will be women with epithelial ovarian, fallopian tube, or primary peritoneal carcinoma after chemotherapy with no evidence of disease or minimal residual disease for primary or recurrent disease; this may or may not be measurable; these patients would normally enter a period of observation after standard management
* Any human leukocyte antigen (HLA) type; (historic HLA typing is permitted)
* Tumor expression of NY-ESO-1 or LAGE-1 by immunohistochemistry (IHC) and/or reverse transcriptase polymerase chain reaction (RTPCR)
* Life expectancy \> 6 months
* Absolute neutrophil count (ANC) \>= 1,000/uL
* Platelets (PLT) \>= 100,000/uL
* Hemoglobin (Hgb) \>= 8 g/dL
* Total bilirubin =\< 1.5 x upper limit of normal (ULN)
* Serum aspartate aminotransferase (serum glutamic oxaloacetic transaminase \[SGOT\]/AST) or serum alanine aminotransferase (serum glutamate pyruvate transaminase \[SGPT\]/ALT) =\< 3 x ULN
* Serum creatinine =\< 2 x ULN
* Have been informed of other treatment options
* Patient or legal representative must understand the investigational nature of this study and sign an Independent Ethics Committee/Institutional Review Board approved written informed consent form prior to receiving any study related procedure
* Have an Eastern Cooperative Oncology Group (ECOG) performance status of =\< 2
* The ability to swallow and retain oral medication
* Patients of child-bearing potential must agree to use acceptable contraceptive methods (e.g., double barrier) during treatment
* Patients may have received previous NY-ESO-1 vaccine therapy; patients who received maintenance paclitaxel or bevacizumab are eligible for enrollment provided they have discontinued therapy (at least 4 weeks for prior taxane or prior bevacizumab) prior to randomization and recovered from toxicities to less than grade 2

Exclusion Criteria:

* Metastatic disease to the central nervous system for which other therapeutic options, including radiotherapy, may be available
* Other serious illnesses (e.g., serious infections requiring antibiotics, bleeding disorders)
* History of severe autoimmune disorders requiring use of steroids or other immunosuppressives
* Concomitant systemic treatment with chronic use (based on the investigator's judgment) of corticosteroids, anti-histamine or non-steroidal anti-inflammatory drugs, and other platelet inhibitory agents
* Chemotherapy, radiation therapy, or immunotherapy within 4 weeks prior to first dosing of study drug (6 weeks for nitrosoureas); concomitant hormonal therapies for breast cancers are allowed
* Subjects being treated with a monoamine oxidase inhibitor (MAOI), or drug which has significant MAOI activity (e.g., Meperidine, linezolid, methylene blue) within 3 weeks prior to screening
* Subjects who are currently receiving therapy with a potent cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) inducer or inhibitor (e.g. clarithromycin, telithromycin, nefazodone, itraconazole, ketoconazole, atazanavir)
* Use of UDP glucuronosyltransferase 1 family, polypeptide A9 (UGT1A9) inhibitor including: diclofenac, imipramine, and ketoconazole
* Participation in any other clinical trial involving another investigational agent within 4 weeks prior to first dosing of study drug
* Known hepatitis B, hepatitis C, or human immunodeficiency virus (HIV)
* Mental impairment that may compromise the ability to give informed consent and comply with the requirements of the study
* Lack of availability of a patient for immunological and clinical follow-up assessment
* Evidence of current drug or alcohol abuse or psychiatric impairment, which in the Investigator's opinion will prevent completion of the protocol therapy or follow-up
* Pregnant or nursing female patients
* Unwilling or unable to follow protocol requirements
* Any condition which in the Investigator's opinion deems the patient an unsuitable candidate to receive study drug (i.e., any significant medical illness or abnormal laboratory finding that would, in the investigator's judgment, increase the patient's risk by participating in this study)
* Known hypersensitivity to any of the study drugs that will be given to the participant
* Additional exclusion criteria for exploratory cohort ONLY: Known pulmonary hypertension