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/…