Ibrutinib and Rituximab in Treating Patients With Relapsed or Refractory Mantle Cell Lymphoma or Older Patients With Newly Diagnosed Mantle Cell Lymphoma

Inclusion Criteria: * Relapsed/refractory MCL: Confirmed diagnosis of mantle cell lymphoma with cluster of differentiation (CD)20 and cyclin D1 through cyclin D3 positivity in tissue biopsy * Relapsed/refractory MCL: Patient has relapsed and or refractory MCL and must have received at least one prior treatment regimen for their disease; patient with leukemia phase (peripheral blood involvement), leptomeningeal disease, cerebral spinal fluid (CSF) MCL, central nervous system (CNS) MCL, non-measurable disease, gastrointestinal (GI) MCL, or bone marrow (BM) MCL are also eligible * Relapsed/refractory MCL: Understand and voluntarily sign an Institutional Review Board (IRB)-approved informed consent form * Relapsed/refractory MCL: Patients with bone marrow or gastrointestinal (GI) only involvement are acceptable * Relapsed/refractory MCL: Eastern Cooperative Oncology Group (ECOG) performance status of 2 or less * Relapsed/refractory MCL: Absolute neutrophil count (ANC) \>= 500/mm\^3; (patients who have bone marrow infiltration by MCL are eligible if their ANC is \>= 500/mm\^3 \[growth factor allowed\]; these patients should be discussed with either the principal investigator \[PI\] or Co-PI of the study for final approval) * Relapsed/refractory MCL: Platelet count \>= 30,000/mm\^3 (transfusion to reach platelet count allowed); (patients who have bone marrow infiltration by MCL are eligible if their platelet level is equal to or \> than 15,000/mm\^3; these patients should be discussed with either the PI or Co-PI of the study for final approval) * Relapsed/refractory MCL: Aspartate transaminase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\]) and alanine transaminase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) \< 2 x upper limit of normal or \< 5 x upper limit of normal if hepatic metastases are present * Relapsed/refractory MCL: Serum bilirubin \< 1.5 mg/dl * Relapsed/refractory MCL: Creatinine (Cr) clearance \>= 30 mL/min * Relapsed/refractory MCL: Patients must be willing to receive transfusions of blood products * Relapsed/refractory MCL: Willing and able to participate in all study related procedures and therapy including swallowing capsules without difficulty * Newly diagnosed MCL: Confirmed diagnosis of MCL with CD20 and cyclin D1 positivity in tissue biopsy; patients must have never received any prior therapy for their disease * Newly diagnosed MCL: Understand and voluntarily sign an IRB-approved informed consent form * Newly diagnosed MCL: Age \> 65 years at the time of signing the informed consent * Newly diagnosed MCL: Patients should in general have bi-dimensional measurable disease with their biggest tumor less than 10 cm; (bone marrow or gastrointestinal \[GI\] only involvement is acceptable) * Newly diagnosed MCL: Eastern Cooperative Oncology Group (ECOG) performance status of 2 or less * Newly diagnosed MCL: Absolute neutrophil count (ANC) \> 750/mm\^3; patients who have bone marrow infiltration by MCL are eligible if their ANC is equal to or \> than 500 * Newly diagnosed MCL: Platelet count \> 50,000/mm\^3; patients who have bone marrow infiltration by MCL are eligible if their platelet level is equal to or \> than 15,000 /mm\^3; (platelet transfusions are allowed; these patients should be discussed with either the PI or Co-PI of the study for final approval) * Newly diagnosed MCL: AST (SGOT) and ALT (SGPT) \< 2 x upper limit of normal or \< 5 x upper limit of normal if hepatic metastases are present * Newly diagnosed MCL: Uric acid within normal limits (allopurinol is allowed to bring abnormal level to within normal limits) * Newly diagnosed MCL: Serum bilirubin \< 1.5 mg/dl * Newly diagnosed MCL: Creatinine (Cr) Clearance \>= 30 mL/min * Newly diagnosed MCL: Ki67 protein (Ki67) \< 50% * Newly diagnosed MCL: Disease free of prior malignancies of equal to or greater than 6 months with exception of currently treated basal cell, squamous cell carcinoma of the skin, carcinoma "in situ" of the cervix or breast, or other malignancies in remission (including prostate cancer patients in remission from radiation therapy, surgery or brachytherapy), not actively being treated, with a life expectancy \> 3 years * Newly diagnosed MCL: Patients must be willing to receive transfusions of blood products * Newly diagnosed MCL: Willing and able to participate in all study related procedures and therapy including swallowing capsules without difficulty Exclusion Criteria: * Relapsed/refractory MCL: Any serious medical condition that places the patient at unacceptable risk and/or would prevent the subject from signing the informed consent form; examples include but are not limited to, uncontrolled hypertension, uncontrolled diabetes mellitus, active/symptomatic coronary artery disease, active infection, active hemorrhage, or psychiatric illness * Relapsed/refractory MCL: Pregnant or breast feeding females * Relapsed/refractory MCL: Known human immunodeficiency virus (HIV) infection; patients with active hepatitis B infection (not including patients with prior hepatitis B vaccination; or positive serum hepatitis B antibody); known hepatitis C infection is allowed as long as there is no active disease and is cleared by gastrointestinal (GI) consultation * Relapsed/refractory MCL: The patient has a prior or concurrent malignancy that in the opinion of the investigator, presents a greater risk to the patient's health and survival, than of the MCL, within the subsequent 6 months at the time of consent * Relapsed/refractory MCL: History of stroke or intracranial hemorrhage within 6 months prior to signing the consent * Relapsed/refractory MCL: Clinically significant cardiovascular disease such as uncontrolled or symptomatic arrhythmias, congestive heart failure or myocardial infarction within 6 months at the time of consent or any class 3 (moderate) or 4 (severe) cardiac disease defined by the New York Heart Association classification * Relapsed/refractory MCL: Significant screening electrocardiogram (ECG) abnormalities including left bundle branch block, 2nd degree atrioventricular (AV) block type II, 3rd degree block, bradycardia (\< 50 beats per minute \[bpm\]), or corrected QT (QTc) \> 500 msec * Relapsed/refractory MCL: Malabsorption syndrome, disease significantly affecting gastrointestinal function, or resection of the stomach or small bowel or ulcerative colitis, symptomatic inflammatory bowel disease, or partial or complete bowel obstruction * Relapsed/refractory MCL: Prior chemotherapy within 3 weeks, nitrosoureas within 6 weeks, therapeutic anticancer antibodies within 4 weeks, radio- or toxin-immunoconjugates within 10 weeks, radiation therapy or other investigational agents within 3 weeks, major surgery within 4 weeks or vaccination with live attenuated vaccines within 4 weeks of the first dose of study drug * Relapsed/refractory MCL: Prior treatment with ibrutinib * Relapsed/refractory MCL: Requires anticoagulation with warfarin or equivalent vitamin K antagonist * Relapsed/refractory MCL: Requires treatment with strong cytochrome P450, family 3, subfamily A, polypeptide 4/5 (CYP3A4/5) inhibitors * Newly Diagnosed MCL: Any serious medical condition that places the patient at unacceptable risk and/or would prevent the subject from signing the informed consent form; examples include but are not limited to, uncontrolled hypertension, uncontrolled diabetes mellitus, active/symptomatic coronary artery disease, active infection requiring treatment with systemic antibiotics, antiviral or antifungal agents, active hemorrhage, or psychiatric illness * Newly diagnosed MCL: Known HIV infection; patients with active hepatitis B infection (not including patients with prior hepatitis B vaccination; or positive serum Hepatitis B antibody); known hepatitis C infection is allowed as long as there is no active disease and is cleared by GI consultation * Newly diagnosed MCL: The patient has a prior or concurrent malignancy that in the opinion of the investigator, presents a greater risk to the patient's health and survival, than of the MCL, within the subsequent 6 months at the time of consent * Newly diagnosed MCL: History of stroke or intracranial hemorrhage within 6 months prior to signing the consent * Newly diagnosed MCL: Clinically significant cardiovascular disease such as uncontrolled or symptomatic arrhythmias, congestive heart failure or myocardial infarction within 6 months at the time of consent or any class 3 (moderate) or 4 (severe) cardiac disease defined by the New York Heart Association Classification * Newly diagnosed MCL: Significant screening electrocardiogram (ECG) abnormalities including left bundle branch block, 2nd degree AV block type II, 3rd degree block, bradycardia (\< 50 bpm), or QTc \> 500 msec * Newly diagnosed MCL: Malabsorption syndrome, disease significantly affecting gastrointestinal function, or resection of the stomach or small bowel or ulcerative colitis, symptomatic inflammatory bowel disease, or partial or complete bowel obstruction * Newly diagnosed MCL: Major surgery within 4 weeks or vaccination with live attenuated vaccines within 4 weeks of the first dose of study drug * Newly diagnosed MCL: Prior treatment with ibrutinib * Newly diagnosed MCL: Requires concomitant anticoagulation with warfarin or equivalent vitamin K antagonist * Newly diagnosed MCL: Requires treatment with strong CYP3A4/5 inhibitors * Newly diagnosed MCL: Patients with blastoid and pleomorphic variants * Newly diagnosed MCL: Ki-67 to be equal or more than 50% * Newly diagnosed MCL: Central nervous system lymphoma * Newly diagnosed MCL: Patients with bi-dimensional measurable disease with a tumor \>= 10 cm