Ixazomib and Dexamethasone With or Without Venetoclax in Treating Patients With Non-t(11;14) Relapsed or Refractory Multiple Myeloma

Inclusion Criteria: * ELIGIBILITY CRITERIA FOR PHASE I AND PHASE II * Patients must have RRMM without t(11;14), confirmed by fluorescence in situ hybridization (FISH). * RRMM with measurable disease with at least one of the following: M-protein \>= 0.5 g/dL in serum or \>= 200 mg/24-hour in urine, or serum free light chain (FLC) \>= 10 mg/dL with abnormal serum FLC ratio for subjects without measurable disease by serum protein electrophoresis (SPEP) or urine protein electrophoresis (UPEP) criteria. * Prior multiple myeloma (MM) treatment: * Received at least 2 lines of prior therapy (including at least one PI, excluding MLN9708), or * Received 1 prior line of therapy with both a PI (excluding MLN9708) and an immunomodulatory (IMiD) agent. * PI-refractory (only applicable to phase 2, cohort 2): progressing =\< 60 days of the last PI therapy or \< 25% response while on therapy. * PI-non-refractory (only applicable to phase 2, cohort 1): not meeting PI-refractory criteria, i.e., (a) did not progress =\< 60 days of the last PI therapy and (b) had \>= 25% response while on therapy. * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%). * Absolute neutrophil count (ANC) \>= 1,000/mcL (patients may not use growth factor support to achieve ANC criteria for eligibility assessment). * Platelets \>= 50,000/mcL (patients may not receive a platelet transfusion within 72 hours of eligibility assessment). * 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 \[SGPT\]) =\< 3.0 x institutional ULN. * Calculated creatinine clearance \>= 30 mL/min (measured by either 24-hour urine collection or calculated using the Cockcroft-Gault formula). * For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated. * Patients with a history of hepatitis C virus (HCV) infection must have completed treatment previously and cured (i.e., undetectable HCV viral load). * Patients with treated brain metastases are eligible if there is no evidence of progression for at least 4 weeks after central nervous system (CNS)- directed treatment, as ascertained by clinical examination and brain imaging (magnetic resonance imaging \[MRI\] or computed tomography \[CT\] scan) during the screening period. * Patients with new or progressive brain metastases (active brain metastases) or leptomeningeal disease are eligible if the treating physician determines that immediate central nervous system (CNS) specific treatment is not required and is unlikely to be required for at least 4 weeks (or scheduled assessment after the first cycle of treatment), and a risk-benefit analysis (discussion) by the patient and the investigator favors participation in the clinical trial. * The effects of MLN9708 and venetoclax in combination with dexamethasone on the developing human fetus are unknown. For this reason and because corticosteroids like dexamethasone are known to be teratogenic, 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, for the duration of study participation, and 3 months after completion of the administration of the study agents. 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. * Ability to understand and the willingness to sign a written informed consent document. * Known human immunodeficiency virus (HIV)-positive patients who meet the following criteria will be considered eligible: * CD4 count \> 350 cells/mm\^3 * Undetectable viral load * Maintained on modern therapeutic regimens (utilizing non-CYP-interactive agents) * No history of acquired immunodeficiency syndrome (AIDS)-defining opportunistic infections. Exclusion Criteria: * Patients who have been previously treated with MLN9708 or venetoclax or other direct BCL2 inhibitor. * Patients who have not recovered from adverse events due to prior anti-cancer therapy (i.e., have residual toxicities grade \> 1), except for grade 2 peripheral sensory neuropathy. * Patients who are receiving any other anti-myeloma chemotherapy or radiotherapy, immunotherapy, or investigational agents within 2 weeks or 5 half-lives (whichever is longer and/or applicable) of the start of the trial. * Patients receiving dexamethasone \>= 40 mg/day or equivalent of any corticosteroids within 2 weeks of the start of the trial. * Patients with non-secretory MM, plasma cell leukemia (i.e., \>= 20% plasma cells in peripheral blood differential or \>= 2,000/mcL circulating plasma cells), symptomatic primary light chain (AL) amyloidosis, POEMS syndrome (polyneuropathy, organomegaly, endocrinopathy, monoclonal protein, and skin changes). * Prior allogeneic hematopoietic cell transplantation (HCT) within the last 12 months and evidence of active graft-versus-host disease (GVHD). * Prior autologous HCT within the last 3 months. * History of active malignancies other than MM within the last 2 years unless treated with curative intent and has no evidence of active disease. Basal cell carcinoma of the skin or localized squamous cell carcinoma of the skin are excluded from this criterion. * History of allergic reactions attributed to compounds of similar chemical or biologic composition to MLN9708, venetoclax, or dexamethasone. This includes boron and boron-containing products. * Patients requiring chronic administration of any medications or substances that are strong inhibitors or inducers of CYP3A4 enzyme are ineligible. Because the lists of these agents are constantly changing, it is important to regularly consult a frequently-updated medical reference. As part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product. * Patients with uncontrolled intercurrent illness. * Female patients who are lactating or have a positive serum pregnancy test during the screening period are excluded from this study because MLN9708 is a proteasome inhibitor with the potential for embryo-lethal effects, and an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with MLN9708. Patients must stop breastfeeding while on MLN9708 and until 90 days have passed since their last dose. These potential risks may also apply to other agents used in this study.