Alisertib, Bortezomib, and Rituximab in Treating Patients With Relapsed or Refractory Mantle Cell Lymphoma or B-cell Low Grade Non-Hodgkin Lymphoma

Inclusion Criteria: * Patients must have histologically confirmed relapsed or refractory mantle cell lymphoma or low grade B-cell non-Hodgkin lymphoma (NHL); patients with evidence of transformation to a high grade histology will not be eligible * Measurable or evaluable disease, as defined in 2008 Revised Response Criteria for Malignant Lymphoma; baseline scans used for measurement should be obtained within 30 days of registration, and baseline bone marrow biopsy and/or aspiration should be obtained with 90 days of registration * Patients may have received one or more lines of prior chemotherapy with/without rituximab (including high dose therapy plus stem cell transplant which is counted as one regimen); prior bortezomib is allowed; patients must not have received bortezomib in the past 6 months * Patients with human immunodeficiency virus (HIV) who are not receiving cytochrome p450 inhibitors, and who have a minimum of 300+ CD4+ cells/mm\^3, an undetectable viral load, and no history of acquired immune deficiency syndrome (AIDS) indicator conditions * Patients must be able to take oral medication and to maintain a fast as required for 2 hours before and 1 hour after MLN8237 administration * Eastern Cooperative Oncology Group (ECOG) performance status 0, 1, or 2 (Karnofsky \>= 60%) * Leukocytes \>= 3,000/mcL (obtained within 30 days of registration) * Absolute neutrophil count \>= 1,500/mcL (obtained within 30 days of registration) * Platelets \>= 75,000/mcL or \>= 50,000/mcL with documented bone marrow involvement (obtained within 30 days of registration) * Total bilirubin within normal institutional limits (may be elevated if direct bilirubin normal) (obtained within 30 days of registration) * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) \< 3 X institutional upper limit of normal (obtained within 30 days of registration) * Creatinine within normal institutional limits OR creatinine clearance \>= 60 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal (obtained within 30 days of registration) * 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 and for the duration of study participation; 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; men treated on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of MLN8237 administration * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C, 3 weeks for rituximab) prior to entering the study or those who have not recovered from adverse events due to agents administered more than 4 weeks earlier * Patients who are receiving any other investigational agents * Patients with known brain metastases should be excluded from this clinical trial * History of allergic reactions attributed to compounds of similar chemical or biologic composition to MLN8237, bortezomib or rituximab * 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 * Known history of uncontrolled sleep apnea syndrome and other conditions that could result in excessive daytime sleepiness, such as severe chronic obstructive pulmonary disease; requirement for supplemental oxygen, or any conditions that could result in excessive toxicity associated with the benzodiazepine-like effects of MLN8237 * Inability to swallow oral medication or to maintain a fast as required for 2 hours before and 1 hour after MLN8237 administration or any condition that would modify small bowel absorption of oral medications, including malabsorption, or resection of pancreas or upper bowel * Pregnant women are excluded from this study; breastfeeding should be discontinued if the mother is treated with MLN8237 * HIV-positive patients on combination antiretroviral therapy which include cytochrome p450 inhibitors are ineligible; patients with CD4 counts less than 300 CD4+ cells/mm\^3 and or a high viral load are ineligible * Grade 2 or greater neuropathy * The following agents are not permitted while patients are taking MLN8237, and should be discontinued at prior to registration if patients are taking them: * Patients must stop using the proton pump inhibitor (PPI) for at least 4 days prior to the first dose of MLN8237; administration of PPI while on study is not permitted * Histamine-2 (H2) receptor antagonists are not permitted from the day prior through to the end of MLN8237 dosing, except as required for premedication for rituximab; constant dosing of H2 blockers is not permitted * Antacid preparations are not permitted for 2 hours before or 2 hours after administration of MLN8237 * Administration of pancreatic enzymes is not permitted at any time while on study * Co-administration of enzyme-inducing antiepileptic drugs, rifampin, rifabutin, rifapentine or St. John's wort is not permitted * Concurrent bisphosphonate therapy is allowed if it was started before study entry and is maintained at recommended dosing intervals; if bisphosphonate therapy is initiated after study entry, bone lesions will not be considered evaluable for disease response * Patients must be willing not drive, operate dangerous tools or machinery, or engage in any other potentially hazardous activity that requires full alertness and coordination if they experience excessive sedation; if a patient experiences excessive sedation believed to be related to MLN8237, treatment with MLN8237 should be interrupted * Patients must be willing to limit alcohol consumption to no more than 1 standard unit of alcohol (12 oz beer \[350 mL\], 1.5 oz \[45 mL\] of 80-proof alcohol, or one 6-oz \[175 mL\] glass of wine) per day during the study and for 30 days from the last dose of MLN8237; minimize the use of agents with central nervous system (CNS) effects * Benzodiazepine use is discouraged but not prohibited