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 contraception prior to the study, for the duration of study participation, and 4 months after completion of MLN8237 plus romidepsin administration * Ability to understand and the willingness to sign a written informed consent document * According to current guidelines, 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; these guidelines may change pending results from an ongoing food effects study Exclusion Criteria: * Patients who have had chemotherapy, radiation therapy, or other investigational agents within 3 weeks prior to entering study, 6 weeks for nitrosoureas or mitomycin * 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, including but not limited to established allergic reaction to benzodiazepines, or romidepsin; agents that alter gastric pH may change MLN8237 absorption are not permitted; proton pump inhibitors need to be stopped 4 days prior to the first dose of MLN8237; histamine-2 (H2) receptor antagonists are not permitted from the day prior (day -1) through to the end of MLN8237 dosing; antacid preparations are not permitted for 2 hours before or 2 hours after administration of MLN8237 * 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; bisphosphonate therapy may not be initiated after study entry * Any serious active disease or co-morbid condition, which in the opinion of the principal investigator, will interfere with the safety or compliance of the trial * Pregnant women are excluded from this study; breastfeeding should be discontinued if the mother is treated with MLN8237 and/or romidepsin * Ejection fraction (EF) \< 40% or myocardial infarction (MI) within the past 3 months; 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 * Requirement for constant administration of proton pump inhibitor, H2 antagonist, or pancreatic enzymes; intermittent uses of antacids or H2 antagonists are allowed * 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; treatment with clinically significant enzyme inducers, such as the enzyme-inducing antiepileptic drugs phenytoin, carbamazepine, oxcarbazepine, primidone or phenobarbital, or rifampin, rifabutin, rifapentine, or St. John's wort within 14 days prior to the first dose of MLN8237 and during the study; patients must be cautiously co-medicated with agents that cause corrected QT interval (QTc) prolongation and agents that are strong or moderate enzyme inhibitors during the study