Pevonedistat and Pembrolizumab for the Treatment of dMMR/MSI-H Metastatic or Locally Advanced Unresectable Solid Tumor

Inclusion Criteria: * Patients 18 years or older * Patients must have metastatic or locally advanced unresectable solid tumor * Tumor that is deficient in mismatch repair (dMMR) or microsatellite instability high (MSI-H) as determined by one of three methods: * Immunohistochemistry determined dMMR by complete loss of MLH1, PMS2, MSH2 or MSH6 * Polymerase chain reaction (PCR) determined microsatellite instability at \> 30% of tested microsatellites * Next-generation determined MSI-H based upon instability at multiple microsatellites as determined by the specific next generation sequencing panel * Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 * Hemoglobin \>= 8 g/dL (may transfuse to achieve this threshold) * Absolute neutrophil count (ANC) \>= 1,500/mm\^3 * Platelet count \>= 100,000/mm\^3 * Albumin \> 2.7 g/dL * Total bilirubin =\< 1.5 x upper limit of normal (ULN) except in patients with Gilbert's syndrome. Patients with Gilbert's syndrome may enroll if direct bilirubin =\< 3 x ULN of the direct bilirubin * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 3.0 x ULN * Creatinine clearance \>= 30 mL/min according to MD Anderson standard, automated laboratory calculation * Human immunodeficiency virus (HIV) patients may be considered as long as they meet the following criteria: * CD4 count \> 350 cells/mm\^3 * Undetectable viral load * No history of acquired immunodeficiency syndrome (AIDS)-defining opportunistic infections * Female patients who: * Are postmenopausal for at least 1 year before the screening visit, OR * Are surgically sterile, OR If they are of childbearing potential: * Agree to practice 1 highly effective method and 1 additional effective (barrier) method of contraception, at the same time, from the time of signing the informed consent through 4 months after the last dose of study drug (female and male condoms should not be used together), or * Agree to practice true abstinence, when this is in line with the preferred and usual lifestyle of the subject. (Periodic abstinence \[eg, calendar, ovulation, symptothermal, post-ovulation methods\] withdrawal, spermicides only, and lactational amenorrhea are not acceptable methods of contraception.) * Male patients, even if surgically sterilized (i.e., status post-vasectomy), who: * Agree to practice effective barrier contraception during the entire study treatment period and through 4 months after the last dose of study drug (female and male condoms should not be used together), or * Agree to practice true abstinence, when this is in line with the preferred and usual lifestyle of the subject. (Periodic abstinence \[e.g., calendar, ovulation, symptothermal, postovulation methods for the female partner\] withdrawal, spermicides only, and lactational amenorrhea are not acceptable methods of contraception.) * Voluntary written consent must be given before performance of any study related procedure not part of standard medical care, with the understanding that consent may be withdrawn by the patient at any time without prejudice to future medical care * Demonstrated prior progression on or after anti-PD1/L1 based therapy by radiographic progression. The potential for psuedoprogression should be excluded by concurrent tumor marker (for example carcinoembryonic antigen) or circulating tumor deoxyribonucleic acid \[ctDNA\] elevation, or clinical symptom progression, or short interval repeat imaging confirming progression. (if uncertain if patient meets eligibility please discuss with study principal investigator \[PI\]) * Must have received at least 2 doses of a PD1/PD-L1 inhibitor * Progressive disease either during therapy or within 2 months of last dose of therapy * Measurable disease by immune modified Response Evaluation Criteria in Solid Tumors (iRECIST) version (v)1.1 by treating investigator or study PI * Tumor that is accessible to biopsy and patient is willing to undergone mandatory tumor biopsies at pre-treatment and on-treatment (unless exception granted by study PI) * Life expectancy \>= 12 weeks as judged by treating physician Exclusion Criteria: * Treatment with any investigational products within 4 weeks before the first dose of any study drug * Any serious medical or psychiatric illness that could, in the investigator's opinion, potentially interfere with the completion of study procedures * Active uncontrolled infection or severe infectious disease, such as severe pneumonia, meningitis, or septicemia * Major surgery within 14 days before the first dose of any study drug or a scheduled surgery during study period * Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial * Life-threatening illness unrelated to cancer * Patients with uncontrolled coagulopathy or bleeding disorder * Known hepatitis B surface antigen seropositive or known or suspected active hepatitis C infection * Note: Patients who have isolated positive hepatitis B core antibody (i.e., in the setting of negative hepatitis B surface antigen and negative hepatitis B surface antibody) must have an undetectable hepatitis B viral load. Patients who have positive hepatitis C antibody may be included if they have an undetectable hepatitis C viral load * Known hepatic cirrhosis or severe pre-existing hepatic impairment * Known cardiopulmonary disease defined as: * Unstable angina; * Congestive heart failure (New York Heart Association \[NYHA\] Class III or IV); * Myocardial infarction (MI) within 6 months prior to first dose (patients who had ischemic heart disease such as a (ACS), MI, and/or revascularization greater than 6 months before screening and who are without cardiac symptoms may enroll); * Symptomatic cardiomyopathy; * Clinically significant arrhythmia: * History of polymorphic ventricular fibrillation or torsade de pointes, * Permanent atrial fibrillation \[a fib\], defined as continuous a fib for \>= 6 months, * Persistent a fib, defined as sustained a fib lasting \> 7 days and/or requiring cardioversion in the 4 weeks before screening, * Grade 3 a fib, within 6 months of starting protocol therapy, defined as symptomatic and incompletely controlled medically, and * Patients with paroxysmal a fib or \< Gr 3 a fib for period of at least 6 months are permitted to enroll provided that their rate is controlled on a stable regimen. * Clinically significant pulmonary hypertension requiring pharmacologic therapy * Uncontrolled high blood pressure (i.e., systolic blood pressure \>= 160 mm Hg, diastolic blood pressure \>= 100 mm Hg) * Prolonged rate corrected QT (QTc) interval \>= 500 msec, calculated according to institutional guidelines * Left ventricular ejection fraction (LVEF) \< 50% as assessed by echocardiogram or radionuclide angiography * Known moderate to severe chronic obstructive pulmonary disease, interstitial lung disease, and pulmonary fibrosis * Known central nervous system (CNS) involvement unless controlled with surgery or radiation therapy and has demonstrated no progression over 3 months from last local modality therapy * Systemic antineoplastic therapy or radiotherapy for other malignant conditions within 14 days before the first dose of any study drug * Female patients who are both lactating and breastfeeding or have a positive serum pregnancy test during the screening period or a positive urine pregnancy test on day 1 before first dose of study drug * Female patients who intend to donate eggs (ova) during the course of this study or 4 months after receiving their last dose of study drug(s) * Male patients who intend to donate sperm during the course of this study or 4 months after receiving their last dose of study drug(s) * Known hypersensitivity to a study agent(s) * Residual adverse events from prior therapy (other than endocrinopathies or alopecia or neuropathy due to chemotherapy) that have not resolved to grade 0-1 * Serious adverse immune related adverse events (grade 3 or 4) with previous immune checkpoint therapy, that were symptomatic and required prolong immunosuppression (\> 6 weeks)