Tacrolimus, Nivolumab, and Ipilimumab in Treating Kidney Transplant Recipients With Selected Unresectable or Metastatic Cancers

Inclusion Criteria: * Patients must be kidney transplant recipients with a functioning allograft who do not currently require dialysis * Patients must have histologically or cytologically confirmed melanoma, basal cell carcinoma, Merkel cell carcinoma, cutaneous squamous cell carcinoma, or microsatellite instability (MSI)-high cancers for which standard non-immunological medical, surgical, or radiation therapy would be insufficient (i.e., patients who are not surgical candidates). This trial is not intended to provide therapy as a neoadjuvant approach * Measurable disease as defined by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 criteria, i.e., 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 by chest x-ray or as \>= 10 mm with computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam is preferred, but not required. Patients with evaluable disease but no target lesions (e.g., evaluable bone metastases) may be included after discussion with the principal investigator (PI) * Patients must have documentation, in consultation with the PI, that they received, refused, or were ineligible for the following non-immunologic therapies: * For patients with: * BRAF-mutant melanoma: prior therapies include BRAF/MEK inhibitors * Merkel cell carcinoma: prior therapies include platinum + VP-16 * Basal cell carcinoma: prior therapies include Hedgehog pathway inhibitors * Cutaneous squamous cell carcinoma: prior therapies include cetuximab * MSI colorectal carcinoma: prior therapies include: leucovorin calcium, 5-fluorouracil and oxaliplatin (FOLFOX) * Patients must have Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Leukocytes \>= 2,000/mcL * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 100,000/mcL * Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN) * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional ULN * Serum creatinine =\< 3 x ULN * Note: patients with creatinine levels above 3 x ULN may be eligible after consultation with the study PI * The effects of nivolumab and ipilimumab on the developing human fetus are unknown. For this reason, and because other therapeutic agents used in this trial are known to be teratogenic, women of child-bearing potential (WOCBP) and men must agree to use adequate contraception (e.g., hormonal or barrier methods of birth control, or abstinence) prior to study entry, for the duration of study participation, and for 31 weeks after the last dose of nivolumab or ipilimumab. Women who are not of childbearing potential (i.e., who are postmenopausal or surgically sterile) as well as azoospermic men do not require contraception. WOCBP must have a negative serum or urine pregnancy test (minimum sensitivity 25 IU/L or equivalent units of beta-human chorionic gonadotropin \[B-HCG\]) during the screening period. Follow-up evaluations will include interval sexual/menstrual histories as needed. Men who receive nivolumab or ipilimumab and are sexually active with WOCBP must use a contraceptive method with a failure rate of \< 1% per year for the duration of the study and for a period of 7 months after the last dose of nivolumab or ipilimumab. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she (or the participating partner) should inform the treating physician immediately. WOCBP is defined as any female who has experienced menarche and who has not undergone surgical sterilization (hysterectomy or bilateral oophorectomy) or who is not postmenopausal. Menopause is defined clinically as 12 months of amenorrhea in a woman over 45 in the absence of other biological or physiological causes. Women under the age of 55 must have a documented serum follicle stimulating hormone (FSH) level less than 40 mIU/mL to be considered postmenopausal * Human immunodeficiency virus (HIV)-infected patients will be eligible for this trial if they are on effective antiretroviral regimens utilizing non-CYP-interactive agents and have an undetectable viral load. If there is evidence of chronic hepatitis B virus (HBV) infection, HBV viral load must be undetectable on suppressive therapy, if indicated. If there is history of hepatitis C virus (HCV) infection, the patient must have been treated and have undetectable HCV viral load * Patients must be able to understand and be willing to sign a written informed consent document Exclusion Criteria: * Patients must not have received a liver, lung, heart, or pancreas transplant; or allogeneic stem cell transplant; or any kind of bone marrow transplant * Patients must not be unwilling or unable to undergo dialysis * Patients must not have prior evidence of human leukocyte antigen (HLA) or non-HLA donor-specific antibodies (DSA). Patients with detectable DSA but negative dd-cfDNA may be eligible after consultation with the study PI * Patients must not have a history of antibody- or cell-mediated allograft rejection within 3 months of study entry * Patients must not have had chemotherapy or radiotherapy within 4 weeks of study entry or those who have not recovered from adverse events (AEs) due to agents administered more than 4 weeks earlier * Patients must not have had prior treatment for their current cancer with an anti-PD-1, anti-PD-L1, anti-PD-L2, or anti-CTLA-4 antibody, or any other antibody or drug specifically targeting T-cell co-stimulation or immune checkpoint pathways * Patients must not be receiving any other investigational agents * Patients must not have known central nervous system (CNS) metastases or leptomeningeal metastases because of poor prognosis and concerns regarding progressive neurologic dysfunction that would confound the evaluation of neurologic and other AEs. Patients with brain metastases are permitted to enroll if metastases have been treated and there is no MRI evidence of progression for 4 weeks after treatment is complete and no evidence of progression within 28 days prior to study entry * Patients must not have a history of severe hypersensitivity reaction to any monoclonal antibody * Patients must not have a history of allergic reactions attributed to compounds of similar chemical or biologic composition to other agents used in study * Patients must not have uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Pregnant women are excluded from this study because nivolumab and ipilimumab have the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for AEs in nursing infants secondary to treatment of the mother, breastfeeding should be discontinued if the mother is treated with nivolumab or ipilimumab. These potential risks may also apply to other agents used in this study * Patients must not have active autoimmune disease, or history of autoimmune disease that might recur, which may affect vital organ function, and will only be eligible after consultation with the study PI * This includes but is not limited to: * Immune-related neurologic disease, * Multiple sclerosis, * Autoimmune (demyelinating) neuropathy, * Guillain-Barre (GB) syndrome, * Myasthenia gravis, * Systemic autoimmune diseases such as systemic lupus erythematosus (SLE), * Connective tissue diseases, * Scleroderma, * Inflammatory bowel disease (IBD; e.g., ulcerative colitis or Crohn's disease), * Rheumatoid arthritis, and * Sjogren's syndrome * Patients must not have had evidence of active or acute diverticulitis, intra-abdominal abscess, gastrointestinal (GI) obstruction and abdominal carcinomatosis which are known risk factors for bowel perforation should be evaluated for the potential need for additional treatment before coming on study