Selinexor and Ibrutinib in Treating Patients With Relapsed or Refractory Chronic Lymphocytic Leukemia or Aggressive Non-Hodgkin Lymphoma

Inclusion Criteria: * A histologically confirmed diagnosis of CLL according the International Workshop on CLL/SLL/B-cell PLL or variant of these (IWCLL or World Health Organization \[WHO\] Criteria) and meet criteria for treatment or have need for cytoreduction for stem cell transplantation or alternative cell therapy; OR * A histologically confirmed diagnosis of mantle cell lymphoma (MCL) or diffuse large B-cell lymphoma (DLBCL) de novo or in the setting of transformation from an indolent lymphoma (including DLBCL not otherwise specified) according to the World Health Organization criteria for diagnosis of NHL; AND * Patients must have received at least one prior therapy for CLL or NHL, need additional treatment, and meet criteria for relapsed or refractory disease; they may not be a candidate for curative therapy; relapsed disease is defined as a patient who previously achieved a complete remission (CR) or a partial remission (PR), but after a period of six or more months demonstrates evidence of disease progression; refractory disease is defined as progression within six months of the last anti-leukemic or anti-lymphoma therapy, or any response less than a CR or PR * Patients must have an Eastern Cooperative Oncology Group (ECOG) performance status of =\< 2 * Patients with NHL must have objective, documented evidence of disease prior to study entry * Platelet count \>= 50,000/mm\^3 in the absence of bone marrow involvement; patients with bone marrow involvement only require a platelet count of 30,000/mm\^3 * Absolute neutrophil count \>= 1000/mm\^3 in the absence of bone marrow involvement * Creatinine clearance (as calculated by Cockroft Gault equation = \[140-age\] \* mass \[kg\]/\[72 \* creatinine mg/dL) \>= 30mL/min * Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) \< 2.5 times upper limit of normal (ULN) * Total bilirubin =\< 2.0 x ULN * Female patients capable of reproduction and male patients who have partners capable of reproduction must agree to use an effective contraceptive method during the course of the study and for 2 months following the completion of their last treatment * Female of childbearing potential must have a negative serum beta-human chorionic gonadotropin (HCG) pregnancy test result within 3 days of first study dose; female patients who are surgically sterilized or who are \> 45 years old and have not experienced menses for \> 2 years may have beta-HCG pregnancy test waived * Patients who are hepatitis B polymerase chain reaction (PCR) negative who have a recent (\< 6 month) history of intravenous immunoglobulin (IVIG) therapy are eligible; patients with a history of hepatitis B (surface antigen or core antibody positive and PCR positive) must take lamivudine or equivalent drug during study therapy and for one year after completion of all therapy; patients on IVIG who are core antibody positive but PCR negative are not mandated to take prophylaxis * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Patients who are concurrently receiving any other investigational agents * Patients who have received: * Radiation or chemotherapy =\< 4 weeks * Mitomycin C, nitrosureas, or radio-immunotherapy =\< 6 weeks, or * Immunotherapy or targeted therapy (such as kinase inhibitors) =\< 2 weeks prior to cycle 1 day 1(except patients already on ibrutinib) * Palliative steroids for disease related symptoms are allowed as long as dose is tapered down to an equivalent of =\< 10 mg of oral prednisone daily on cycle 1 day 1 * Patients who have underwent autologous or allogeneic stem cell transplant =\< 4 weeks prior to cycle 1 day 1 or have active graft-versus-host disease are excluded * Patients unable to swallow capsules, those with uncontrolled vomiting or diarrhea or disease significantly affecting gastrointestinal function and/or inhibiting small intestine absorption such as: malabsorption syndrome, resection of the small bowel, or poorly controlled inflammatory bowel disease affecting the small intestine * Patients who are 20% below their ideal body weight * Patients must not be receiving systemic anticoagulation with warfarin; patients must be off warfarin for 30 days prior to enrollment; patients who require anticoagulation with an agent other than warfarin will not be excluded, but must be reviewed by the principal investigator prior to enrollment * As ibrutinib is extensively metabolized by cytochrome P450, family 3, subfamily A, polypeptide 4/5 (CYP3A4/5), and patients must not require continued therapy with a strong inhibitor or inducer of CYP3A4/5 * Patients with active human immunodeficiency virus (HIV) or hepatitis B or C * Patients with secondary malignancy that requires active systemic therapy that will interfere with interpretation of efficacy or toxicity of selinexor; (Note: patients with basal or squamous skin carcinoma, cervical carcinoma in situ, localized breast cancer requiring hormonal therapy or localized prostate cancer (Gleason score \< 5 are allowed) * Patients with active known central nervous system (CNS) involvement of CLL or lymphoma; (patients with history of CNS CLL or lymphoma now in remission are eligible for the trial) * Patients who are pregnant or breast feeding; breastfeeding should be discontinued if the mother is treated with selinexor * Patients must have recovered all toxicities from prior therapy or radiation to grade 1 or less (excluding alopecia) * Patients may not have had major surgery within 10 days of enrollment, or minor surgery within 7 days of enrollment; examples of minor surgery include dental surgery, insertion of a venous access device, skin biopsy, or aspiration of a joint; the decision about whether a surgery is major or minor can be made at the discretion of the treating physician * Patients with 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 * Patients with markedly decreased visual acuity