Inclusion Criteria:

* Patients must have histologically or cytologically confirmed relapsed/refractory non-Hodgkin lymphoma (NHL) having progressed after a minimum of one systemic therapy with any of the following T-cell histologies:

* Peripheral T-cell NHL (PTCL) not otherwise specified (NOS)
* Anaplastic large cell T-cell lymphoma (ALCL) that is anaplastic lymphoma kinase either positive or negative
* Angioimmunoblastic T-cell NHL
* Subcutaneous panniculitis-like T-cell lymphoma
* Enteropathy-associated T-cell NHL
* Hepatosplenic T-cell lymphomas
* Extranodal natural killer (NK)/T-cell lymphoma, nasal type
* Adult T-cell leukemia/lymphoma
* Unclassifiable PTCL
* Transformed cutaneous T-cell lymphoma (CTCL) to PTCL with systemic involvement (not local skin transformation)
* No other histologies are eligible; examples of ineligible histologies include: T-cell prolymphocytic leukemia, T-cell large granular lymphocytic leukemia, NK-cell leukemia, mycosis fungoides, Sezary syndrome, lymphomatoid papulosis, and primary CTCL
* Patients must have received at least one course of prior systemic therapy which may include chemotherapy, antibody therapy, or immunotherapy; for all forms of systemic therapy, patients must have completed therapy at least 21 days prior to registration; patients must not be within 84 days of radioimmunotherapy; steroids at a low dose for control of itching (up to the equivalent of 20 mg of prednisone daily) are allowed
* Patients may have received prior radiation in combination with systemic therapy; patients must not be within 21 days of external beam radiation therapy
* Patients must not have received a previous allogeneic stem cell transplant or be within 90 days of an autologous stem cell transplant
* Adequate sections and a paraffin block from the relapsed/refractory specimen must be submitted for review by the lymphoma pathology group; an adequate biopsy requires sufficient tissue to establish the architecture and a Revised European American Lymphoma (REAL) or World Health Organization (WHO) histologic subtype with certainty; thus, core biopsies, especially multiple core biopsies MAY be adequate; whereas, needle aspirations or cytologies are not adequate
* Patients must have bidimensionally measurable disease within 28 days prior to registration; a diagnostic quality computed tomography (CT) scan of the chest abdomen, pelvis, neck and positron emission tomography (PET)/CT must be performed within 28 days of registration (PET/CT scan can be done instead of separate PET and CT scans only if the CT component is a diagnostic CT with contrast); patients who also have non-measurable disease in addition to measurable disease must have all non-measurable disease assessed within 42 days prior to registration
* Patients must have a bilateral or unilateral bone marrow aspirate and biopsy performed within 42 days prior to registration
* Patients must not have clinical evidence of central nervous system involvement by lymphoma; any laboratory tests that are performed to assess clinical signs of central nervous system involvement must have been performed within 42 days prior to registration, and the results must be negative
* Patients must be able to swallow tablets
* Patients known to be human immunodeficiency virus (HIV)-positive must not have multi-drug resistant HIV infection, CD4 counts < 150/mcL, or other concurrent acquired immunodeficiency syndrome (AIDS)-defining conditions
* Patients must be offered the opportunity to consent to the banking of specimens for future use
* Absolute granulocyte count >= 1,500 cells/mcL; patients with documented marrow involvement may be transfused to this value
* Platelet count >= 75,000 cells/mcL; patients with documented marrow involvement may be transfused to this value
* Serum creatinine (mg/dL) =< institutional upper limit of normal (IULN) obtained within 14 days prior to registration
* Calculated creatinine clearance > 50 ml/min; the serum creatinine value used in the calculation must have been obtained within 14 days prior to registration
* Serum bilirubin =< 2 times institutional upper limit of normal
* Serum aspartate aminotransferase (AST) and alanine aminotransferase (ALT) both =< 2.5 x IULN
* Serum lactate dehydrogenase (LDH) obtained within 14 days prior to registration
* Patients must have a Zubrod performance status of 0, 1, or 2
* Patients must NOT have New York Heart Association (NYHA) class II-IV heart failure
* No other prior malignancy is allowed except for the following: adequately treated basal cell or squamous cell skin cancer, in situ cervical cancer, adequately treated stage I or II cancer from which the patient is currently in complete remission, or any other cancer from which the patient has been disease free for five years
* Pregnant or nursing women are not eligible; women/men of reproductive potential 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 and for 4 months after completion of MLN8237 administration; should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, the patient should inform the treating physician immediately
* All patients must be informed of the investigational nature of this study and must sign and give written informed consent in accordance with institutional and federal guidelines