Ixazomib and Pevonedistat in Treating Patients With Multiple Myeloma That Has Come Back or Does Not Respond to Treatment

Inclusion Criteria: * Patients must have RRMM with measurable disease, as defined by at least one of the following: * Serum monoclonal protein \>= 0.5 g/dL * Urinary monoclonal protein excretion of \>= 200 mg/24 hours * Kappa or lambda light chain level \>= 10 mg/dL with an abnormal free light chain ratio * At least two prior lines of therapy and all patients should have at least been exposed to a proteasome inhibitor (PI), an immunomodulatory drug (IMiD), and an anti-CD38 antibody. * For proteasome-sensitive expansion cohort: Patients with MM who relapsed or are refractory to a prior line of therapy not including a proteasome inhibitor * For proteasome-relapsed/refractory expansion cohort: Patients with MM who have relapsed after prior PI exposure or are PI-refractory, defined as nonresponsive to treatment or progresses within 60 days of last exposure to a PI * Age \>= 18 years * Because no dosing or adverse event (AE) data are currently available on the use of MLN4924 (pevonedistat) in combination with MLN9708 (ixazomib) in patients \< 18 years of age, and as this disease is exceptionally uncommon in this age group, children are excluded from this study * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,000/mcL * Platelets \>= 75,000/mcL * Bilirubin =\< institutional upper limit of normal (ULN). * Patients with Gilbert's syndrome may enroll if direct bilirubin =\< 1.5 x ULN * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3.0 x institutional ULN * Creatinine clearance (CrCl) by Cockcroft-Gault \>= 30 mL/min * Known human immunodeficiency virus (HIV) positive patients who meet the following criteria will be considered eligible: * CD 4 count \> 350 cells/mm\^3 * Undetectable viral load * Maintained on modern therapeutic regimens utilizing non-CYP-interactive agents (e.g. excluding ritonavir) * No history of acquired immune deficiency syndrome (AIDS)-defining opportunistic infections * The effects of MLN4924 (pevonedistat) and MLN9708 (ixazomib) on the developing human fetus are unknown. For this reason and because NAE inhibitory agents are known to be teratogenic, women of child-bearing potential and men must meet the following criteria: * Female patients who are: * Postmenopausal for at least one year before the screening visit, OR * Surgically sterile, OR * If of childbearing potential, agree to practice 1 highly effective method and 1 additional (barrier) method of contraception, at the same time, from the time of signing the informed consent until 4 months after the last dose of the ixazomib and pevonedistat (female and male condoms should not be used together), or agree to abstain from heterosexual intercourse, when this is in line with the preferred and usual lifestyle of the subject (Periodic abstinence \[e.g,, calendar, ovulation, symptothermal, postovulation methods\] withdrawal, spermicides only, and lactational amenorrhea are not acceptable methods of contraception) * Male patients, even if surgically sterilized, who: * Agree to practice effective barrier contraception during the entire time enrolled on study through 4 months after completion of ixazomib and pevonedistat administration (female and male condoms should not be used together), OR * Agree to abstain from heterosexual intercourse, 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) * 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. Participants with impaired decision-making capacity (IDMC) who have a legally-authorized representative (LAR) and/or family member available will also be eligible Exclusion Criteria: * Diagnosed or treated for another malignancy within 2 years before randomization or previously diagnosed with another malignancy and have any evidence of residual disease. Patients with non-melanoma skin cancer or carcinoma in situ of any type are not excluded if they have undergone resection * Patients who are receiving any other investigational agents, within 30 days of the start of this trial and throughout the duration of this trial * Patients with known central nervous system involvement should be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other AEs * History of allergic reactions attributed to compounds of similar chemical or biologic composition to MLN4924 (pevonedistat) or MLN9708 (ixazomib) (including boron or boron-containing products) * Patients with uncontrolled intercurrent illness * Pregnant women are excluded from this study because MLN4924 (pevonedistat) is an NAE inhibitory agent with 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 with MLN4924 (pevonedistat), breastfeeding should be discontinued if the mother is treated with MLN4924 (pevonedistat). These potential risks may also apply to the use of MLN9708 (ixazomib) in this study * Major surgery within 14 days before the first dose of any study drug or a scheduled surgery during study period * Patients with uncontrolled coagulopathy or bleeding disorder * Known hepatic impairment as defined by known hepatic cirrhosis, 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 cardiopulmonary disease defined as: * Unstable angina; * Congestive heart failure (New York Heart Association \[NYHA\] class III or IV); * Myocardial infarction within 6 months prior to first dose (patients who had ischemic heart disease such as acute coronary syndrome \[ACS\], myocardial infarction, 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, defined as continuous atrial fibrillation for \>= 6 months, * Persistent atrial fibrillation, defined as sustained atrial fibrillation lasting \> 7 days and/or requiring cardioversion in the 4 weeks before screening, * Grade 3 atrial fibrillation defined as symptomatic and incompletely controlled medically, or controlled with device (e.g., pacemaker), or ablation in the past 6 months and * Patients with paroxysmal atrial fibrillation or grade \< 3 atrial fibrillation 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 \> 180 mmHg, diastolic blood pressure \> 95 mmHg) * Prolonged rate corrected QT (QTc) interval \>= 500 msec, calculated according to institutional guidelines * Left ventricular ejection fraction (LVEF) \< 50% as assessed by echocardiogram * Known moderate to severe chronic obstructive pulmonary disease, interstitial lung disease, and pulmonary fibrosis * Known gastrointestinal (GI) disease or GI procedure that could interfere with the oral absorption or tolerance of MLN9708 (ixazomib), including difficulty swallowing * Peripheral neuropathy that is grade \>= 3, or grade 2 with pain on clinical examination during the screening period * Patients that have previously been treated with MLN9708 (ixazomib) * Systemic treatment, within 14 days before the first dose of MLN9708 (ixazomib), with strong CYP3A inducers (rifampin, rifapentine, rifabutin, ritonavir, carbamazepine, phenytoin, phenobarbital), or use of St. John's wort. Clinically significant metabolic enzyme inducers are not permitted during this study * Radiotherapy within 14 days before enrollment. If the involved field is small, 7 days will be considered a sufficient interval between treatment and administration of the MLN9708 (ixazomib) * 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)