Ixazomib and Rituximab in Treating Patients With Relapsed or Refractory Mantle Cell Lymphoma

Inclusion Criteria: * Patients must have histologically confirmed diagnosis of mantle cell lymphoma * Patients must have measurable disease, as defined by at least one of the following: * Lymph node or mass 2 cm or greater, splenomegaly \> 13 cm * Bone marrow only disease as per morphology or flow cytometry * Patients must have relapsed and/or refractory disease to at least 2 lines of therapy including either an anthracycline- or bendamustine- based regimen and a BTK inhibitor * Patients must have an Eastern Cooperative Oncology Group (ECOG) performance status and/or other performance status 0, 1, or 2 * Absolute neutrophil count (ANC) \>= 1,000/mm\^3 * Platelets \>= 50,000/mm\^3 * Total bilirubin \< 1.5 x institutional upper limit of normal (ULN). In patients with documented Gilbert's syndrome, total bilirubin =\< 2.5 x ULN * Aspartate aminotransferase (AST)/serum glutamic-oxaloacetic transaminase (SGOT) and alanine aminotransferase (ALT)/serum glutamic-pyruvic transaminase (SGPT) =\< 3 x ULN * Creatinine clearance \>= 30 mL/min * Patients must be willing to give written consent before performance of any study related procedures not part of standard medical care, with the understanding that consent may be withdrawn by the patient at any time without prejudice to future medical care * Female patients who: * Are postmenopausal for at least 1 year before the screening visit,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 2 effective methods of contraception, at the same time, from the time of signing the informed consent form through 90 days after the last dose of study drug * Male patients, even if surgically sterilized (i.e., status post-vasectomy), must agree to one of the following: * Agree to practice effective barrier contraception during the entire study treatment period and through 90 days after the last dose of study drug Exclusion Criteria: * Female patients who are lactating or have a positive serum pregnancy test during the screening period * Failure to have fully recovered (i.e., =\< grade 1 toxicity) from the reversible effects of prior chemotherapy * Major surgery within 14 days before enrollment * 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 ixazomib * Central nervous system involvement * Infection requiring systemic antibiotic therapy or other serious infection within 14 days before study enrollment. Patient may be eligible, if infectious disease specialist approves start of therapy AND subject has completed course of antibiotic therapy * Evidence of current uncontrolled cardiovascular conditions, including uncontrolled hypertension, uncontrolled cardiac arrhythmias, symptomatic congestive heart failure, unstable angina, or myocardial infarction within the past 6 months * Systemic treatment, within 14 days before the first dose of ixazomib, with strong CYP3A inducers (rifampin, rifapentine, rifabutin, carbamazepine, phenytoin, phenobarbital), or use of St. John's wort * Ongoing or active systemic infection, active (deoxyribonucleic acid \[DNA\] polymerase chain reaction \[PCR\] positivity) hepatitis B or C virus infection, or known human immunodeficiency virus (HIV) positive * Any serious medical or psychiatric illness that could, in the investigator's opinion, potentially interfere with the completion of treatment according to this protocol * Known allergy to any of the study medications, their analogues, or excipients in the various formulations of any agent * Known gastrointestinal (GI) disease or GI procedure that could interfere with the oral absorption or tolerance of ixazomib including difficulty swallowing * Diagnosed or treated for another malignancy within 2 years before study enrollment or previously diagnosed with another malignancy and have any evidence of residual disease. Patients with nonmelanoma skin cancer or carcinoma in situ of any type are not excluded if they have undergone complete resection * Patient has \>= grade 3 peripheral neuropathy, or grade 2 with pain on clinical examination during the screening period * Participation in other clinical trials, including those with other investigational agents not included in this trial, within 30 days of the start of this trial and throughout the duration of this trial * Patients that have previously been treated with proteasome inhibitors, or participated in a study with proteasome inhibitors whether treated with proteasome inhibitors or not