Alisertib and Romidepsin in Treating Patients With Relapsed or Refractory B-Cell or T-Cell Lymphomas

Inclusion Criteria: * Patients must have histologically or cytologically confirmed Hodgkin lymphoma, Burkitt's lymphoma, double-hit lymphoma, other c-Myc positive B-cell lymphoma, diffuse large-B cell lymphoma including those patients with history of transformed follicular lymphoma, mantle cell lymphoma, or peripheral T-cell lymphoma * Patients must have at least one 1.5 cm bidimensional measurable lesion * Relapsed or refractory after at least 1 front-line therapy * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Life expectancy of greater than 12 weeks * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 75,000/mcL * Direct bilirubin =\< 1 mg/dL * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal * Creatinine =\< 2 x institutional upper limits of normal * 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 4 months after completion of MLN8237 plus romidepsin administration; 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 or enrolled on this protocol must also agree to use adequate contrac…