Ibrutinib With or Without Bortezomib and Dexamethasone in Treating Patients With Relapsed or Refractory Immunoglobulin Light Chain Amyloidosis

Criteria: * Measurable disease of AL amyloidosis as defined by at least ONE of the following: * Serum monoclonal protein \>= 1.0 by protein electrophoresis * \> 200 mg of monoclonal protein in the urine on 24 hour electrophoresis * Free light chains (abnormal absolute value, ratio and the dFLC \> 5 mg/dL) Inclusion Criteria: * Histological diagnosis of AL amyloidosis as based on detection by polarizing microscopy of green birefringent material in Congo red-stained tissue specimens; the type must have been confirmed unequivocally * Must have had one prior line of systemic therapy for AL; Note: patients who do not achieve at least a PR to frontline therapy in 3 months may be eligible after discussion with study chair * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1 or 2 * Absolute neutrophil count (ANC) \>= 1000/mm\^3 * Platelet count \>= 100,000/mm\^3 or \>= 50,000/mm\^3 if bone marrow involvement independent of transfusion support in either situation * Hemoglobin \>= 8.0 g/dL * Total bilirubin =\< 1.5 x upper limit of normal (ULN) unless due to Gilbert's syndrome or of non-hepatic origin * Aspartate transaminase (AST) =\< 3 x upper limit of normal (ULN) * Alanine aminotransferase (ALT) =\< 3 x ULN * Creatinine =\< 3 mg/dL and creatinine clearance (CrCL) \>= 25 ml/min * Negative serum pregnancy test done =\< 7 days prior to registration, for persons of childbearing potential only * Ability to complete questionnaire(s) by themselves or with assistance * Ability to provide written informed consent * Willing to return to enrolling institution for follow-up (during the Active Monitoring Phase of the study) * Willing to provide bone marrow and blood samples for correlative research purposes Exclusion Criteria: * Any of the following: * Pregnant persons * Nursing persons * Persons of childbearing potential who are unwilling to employ adequate contraception; Note: persons of childbearing potential and persons able to father a child who are sexually active must be practicing a highly effective method of birth control during and after the study consistent with local regulations regarding the use of birth control methods for subjects participating in clinical trials; subjects must agree to not donate sperm during and after the study; for persons of childbearing potential, these restrictions apply for 1 month after the last dose of study drug; for persons able to father a child, these restrictions apply for 3 months after the last dose of study drug * Concomitant high dose corticosteroids (concurrent use of corticosteroids) while on single agent ibrutinib; EXCEPTION: patients may be on chronic steroids (maximum dose 20 mg/day prednisone equivalent) if they are being given for disorders other than amyloid, (i.e., adrenal insufficiency, rheumatoid arthritis, etc) * Active malignancy =\< 3 years prior to registration; EXCEPTIONS: Non-melanotic skin cancer or carcinoma-in-situ of the cervix; Note: if there is a history of prior malignancy, they must not be receiving other specific treatment for their cancer * Co-morbid systemic illnesses or other severe concurrent disease which, in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens * Known history of human immunodeficiency virus (HIV) or active hepatitis C virus or active hepatitis B virus infection or any uncontrolled active systemic infection * Any life-threatening illness, medical condition, or organ system dysfunction which, in the investigator's opinion, could compromise the subject's safety, interfere with the absorption or metabolism of ibrutinib capsules, or put the study outcomes at undue risk * Receiving any other investigational agent which would be considered as a treatment for AL amyloidosis * Vaccinated with live, attenuated vaccines =\< 4 weeks prior to registration * Clinically overt multiple myeloma (i.e. original hypercalcemia, renal failure, anemia, bone lesions \[CRAB\] criteria); Note: extent of marrow plasmacytosis is not prohibitive * Clinically significant cardiovascular disease such as uncontrolled or symptomatic arrhythmias, congestive heart failure, or myocardial infarction =\< 6 months prior to registration, or any class 3 (moderate) or class 4 (severe) cardiac disease as defined by the New York Heart Association Functional Classification * NT-ProBNP \> 8,500 pg/mL * Use of strong and moderate CYP3A inhibitors and inducers =\< 7 days prior to registration * Anticoagulation with warfarin or equivalent vitamin K antagonists (e.g., phenprocoumon); Note: use of low molecular weight heparin (or any anticoagulation agent) is allowed provided there is no history of bleeding (minor or major) =\< 12 months prior to registration; the treating physician should discuss the case with the study chair * History of stroke or intracranial hemorrhage =\< 6 months prior to registration * Known central nervous system metastasis * Major surgery or a wound that has not fully healed =\< 4 weeks prior to registration * Surgery or invasive procedure requiring sutures or staples for closure =\< 7 days prior to registration * Minor procedures (such as a central line placement, needle biopsy, thoracentesis, or paracentesis) =\< 3 days prior to registration