Dasatinib and Ipilimumab in Treating Patients With Gastrointestinal Stromal Tumors or Other Sarcomas That Cannot Be Removed by Surgery or Are Metastatic

Inclusion Criteria: * DOSE ESCALATION COHORT: subjects must have histologically or cytologically confirmed sarcoma that is metastatic or unresectable * DOSE EXPANSION COHORT: subjects must have histologically or cytologically confirmed GIST that is metastatic or unresectable * Patients must have measurable disease per RECIST 1.1, 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 \>= 20 mm with conventional techniques or as \>= 10 mm with spiral computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam * DOSE ESCALATION COHORT: patients must have had at least one prior therapy * DOSE EXPANSION COHORT: GIST patients must have had progression on or have been intolerant to imatinib and sunitinib * Eastern Cooperative Oncology Group (ECOG) performance status =\< 1 (Karnofsky \>= 70%) * Life expectancy of greater than 3 months * Leukocytes \>= 3 K/mcL * Absolute neutrophil count \>= 1.5 K/mcL * Platelets \>= 100 K/mcL * Hemoglobin \>= 8.0 g/dl * Total bilirubin =\< 1.5 x institutional upper limit of normal; note: patients with hyperbilirubinemia clinically consistent with an inherited disorder of bilirubin metabolism (e.g., Gilbert syndrome) will be eligible at the discretion of the treating physician and/or the principal investigator * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal * Creatinine =\< 1.5 x institutional upper limit of normal OR creatinine clearance \>= 60 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal * Women of child-bearing potential and men 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; men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of dasatinib administration * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Patients who have had chemotherapy (non-tyrosine kinase inhibitor \[TKI\]) or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events due to agents administered more than 4 weeks earlier * Patients with a history of prior treatment with ipilimumab or dasatinib * Patients who are receiving any other investigational agents * Patients with known brain metastases are excluded from this clinical trial * History of allergic reactions attributed to compounds of similar chemical or biologic composition to dasatinib and ipilimumab * Patients who require concurrent treatment with any medications or substances that are potent inhibitors or inducers of cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) * Patients who require concurrent treatment with any medications or substances that have significant proarrhythmic potential are ineligible * Patients with any condition (e.g., gastrointestinal tract disease resulting in an inability to take oral medication or a requirement for IV alimentation, prior surgical procedures affecting absorption, or active peptic ulcer disease) that impairs their ability to swallow and retain dasatinib tablets are excluded * Patients may not have any clinically significant cardiovascular disease including the following: * Myocardial infarction or ventricular tachyarrhythmia within 6 months * Prolonged corrected QT interval (QTc) \> 480 msec * Ejection fraction less than 50% * Major conduction abnormality (unless a cardiac pacemaker is present) * Uncontrolled intercurrent illness including, but not limited to, the following: ongoing or active infection; history of significant bleeding disorder, including congenital (von Willebrand's disease) or acquired (anti-factor VIII antibodies) disorders; large pleural effusions; or psychiatric illness/social situations that would limit compliance with study requirements * Pregnant women are excluded from this study; breastfeeding should be discontinued if the mother is treated with dasatinib * Subjects may not have known human immunodeficiency virus (HIV), active hepatitis A, or hepatitis B or C infection * Subjects with any active autoimmune disease or a documented history of autoimmune disease or history of syndrome that required systemic steroids or immunosuppressive medications including but not limited to inflammatory bowel disease, rheumatoid arthritis, autoimmune thyroiditis, autoimmune hepatitis, systemic sclerosis (scleroderma and variants), systemic lupus erythematosus, autoimmune vasculitis, autoimmune neuropathies (e.g. Guillain-Barre syndrome), and multiple sclerosis; patients with vitiligo, asthma and diabetes are NOT excluded; final determination can be left to the discretion of the principal investigator * Subjects may not have ongoing chronic diarrhea * Subjects may not have had prior organ allograft or allogeneic bone marrow transplantation * Subjects may not have any major surgery within 4 weeks * Subjects may not have known current drug or alcohol abuse * Subjects may not have an underlying medical condition that in the opinion of the investigator could adversely affect the ability of the subject to comply with or tolerate study procedures and/or study therapy, or confound the ability to interpret the tolerability of combined administration of dasatinib and ipilimumab in treated subjects * Subjects may not have other active malignancies other than indolent malignancies not requiring active therapy which the investigator determines are unlikely to interfere with treatment and safety analysis