Immune Cell Therapy (CAR-T) for the Treatment of Patients With HIV and B-Cell Non-Hodgkin Lymphoma

Inclusion Criteria: * Participant with age \>= 18 years at the time of consent. Because no dosing or adverse event data are currently available on the use of axicabtagene ciloleucel in participants \< 18 years of age, children are excluded from this study * Participant is able to understand and willing to sign a written informed consent document before any study procedures * Participant must have R/R aggressive B-cell NHL of the following histologies: * Diffuse large B-cell lymphoma (DLBCL, including transformed from indolent histology) * High-grade B-cell lymphoma * Primary mediastinal B-cell lymphoma * Follicular lymphoma, grade 3B * Participant must have been treated with an anthracycline and rituximab (or other CD20-targeted agent) and have R/R disease after at least 2 lines of therapy * At least 2 weeks or 5 half-lives, whichever is shorter, must have elapsed since any prior systemic cancer therapy at the time the subject provides consent * Evaluable disease as either: * Positron emission tomography (PET)-positive disease according to the "Recommendations for Initial Evaluation, Staging, and Response Assessment of Hodgkin and Non-Hodgkin Lymphoma: The Lugano Classification", or * Bone marrow involvement assessed by bone marrow biopsy * Eastern Cooperative Oncology Group ECOG performance status =\< 1 (Karnofsky \>= 60%) * Serum creatinine =\< 1.5 x age-adjusted upper limit of normal (ULN) OR calculated creatinine clearance (Cockcroft and Gault) \> 30 mL/min/1.73 m\^2 (within 4 weeks before enrollment) * Alanine aminotransferase (ALT) =\< 5 x ULN and total bilirubin \< 2.0 mg/dL (or \< 3.0 mg/dL for subjects with Gilbert's syndrome or lymphomatous infiltration of the liver or if taking atazanavir or indinavir (within 4 weeks before enrollment) * Adequate pulmonary function, defined as =\< Common Terminology Criteria for Adverse Events (CTCAE) grade 1 dyspnea and oxygen saturation (SaO2) \>= 92% on room air (within 4 weeks before enrollment) * Adequate cardiac function, defined as left ventricular ejection fraction (LVEF) \>= 40% as assessed by echocardiogram or multiple uptake gated acquisition (MUGA) scan performed within 1 month of determination of eligibility * Absolute neutrophil count: \>= 1,000/mm\^3 (within 4 weeks before enrollment) * Platelets: \>= 75,000/mm\^3 (within 4 weeks before enrollment) * Total bilirubin: =\< 1.5 x institutional upper limit of normal (ULN) (3.0 x ULN for patients with Gilbert syndrome) If, however, the elevated bilirubin is felt to be secondary to antiretroviral therapy, the total bilirubin must be =\< 3.5 mg/dL, provided that the direct bilirubin is normal and the aspartate aminotransferase (AST) and ALT =\< 3 x the upper limit of normal (within 4 weeks before enrollment) * Adequate vascular access for leukapheresis procedure and for administration of the cellular product (either peripheral line or leukapheresis catheter) * Participants who have received previous CD19-targeted therapy must have CD19-positive lymphoma confirmed on a biopsy since completing the prior CD19-targeted therapy * The effects of axicabtagene ciloleucel on the developing human fetus are unknown. For this reason, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) before study entry, for the duration of study participation, and 12 months after the last dose of axicabtagene ciloleucel. 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 who have partners of childbearing potential must agree to use an effective barrier contraceptive method before study entry, for the duration of study participation, and for 1 year after the last dose of axicabtagene ciloleucel * Documentation of HIV-1 infection by means of any one of the following: * Documentation of HIV diagnosis in the medical record by a licensed health care investigator; * Documentation of receipt of antiretroviral therapy (ART) (at least two different medications that do not constitute a prescription for pre-exposure prophylaxis \[PrEP\]) by a licensed health care investigator. Documentation may be a record of an ART prescription in the participant's medical record, a written prescription in the name of the participant for ART, or pill bottles for ART with a label showing the participant's name; * HIV-1 ribonucleic acid (RNA) detection by a licensed HIV-1 RNA assay demonstrating \>1000 RNA copies/mL; * Any federally approved, licensed HIV screening antibody and/or HIV antibody/antigen combination assay confirmed by a second licensed HIV assay such as a HIV-1 Western blot confirmation or HIV rapid multispot antibody differentiation assay. NOTE: A "licensed" assay refers to a U.S. Food and Drug Administration (FDA)-approved assay, which is required for all Investigational New Drug (IND) studies * HIV viral load below 50 copies/mL by FDA-approved assays within 4 weeks prior to registration * A CD4 cell count must be obtained within 4 weeks before enrollment at any U.S. laboratory that has a clinical laboratory improvement amendments (CLIA) certification or its equivalent. Twenty participants will be studied with a goal to enroll a minimum of 6 participants with a CD4 \<100 cells/uL * Participants who have hepatitis C (reactive anti-HCV antibody) and hepatitis B (HBsAg positive and/or anti-HBc-Total positive), may be enrolled, provided total bilirubin is =\< 1.5 x institutional upper limit of normal (ULN), AST (serum glutamic oxaloacetic transaminase \[SGOT\]) and ALT (serum glutamic pyruvic transaminase \[SGPT\]) must be =\< 3 X institutional upper limit of normal, and HBV deoxyribonucleic acid (DNA) \<100 IU/mL (if hepatitis B positive) within 4 weeks before enrollment. There must be no evidence of cirrhosis present * Participants with hepatitis B core antibody positive must be on an antiviral agent to suppress hepatitis B throughout the study and be willing to continue therapy for at least one year after axicabtagene infusion * Participants who are willing to continue ART during leukapheresis, manufacturing and infusion and post infusion of axicabtagene ciloleucel Exclusion Criteria: * Participants felt to have a high prospect of clinically benefiting from autologous transplantation * Participants with central nervous system (CNS)-only involvement by malignancy (note: participants with secondary CNS involvement are allowed on study * Participants with a second prior or concurrent malignancy that, in the opinion of the investigator, has a natural history or treatment course that has the potential to interfere with the safety or efficacy assessment of the investigational regimen * Treatment with alemtuzumab within 6 months before anticipated leukapheresis, or treatment with fludarabine or cladribine within 3 months before anticipated leukapheresis * Participants with uncontrolled systemic fungal, bacterial, viral, or other infection despite appropriate antibiotics or other treatment at the time of enrollment * Presence of acute or chronic graft-versus-host disease * History of any one of the following cardiovascular conditions within the past 6 months: class III or IV heart failure as defined by the New York Heart Association, cardiac angioplasty or stenting, myocardial infarction, unstable angina, or other clinically significant cardiac disease * History or presence of clinically relevant CNS pathology such as epilepsy, seizure, paresis, aphasia, stroke, severe brain injuries, dementia, Parkinson's disease, cerebellar disease, organic brain syndrome, or psychosis * Pregnant or nursing women. NOTE: Women of reproductive potential must have a negative serum pregnancy test performed within 48 hours before starting conditioning chemotherapy. Pregnant women are excluded from this study because axicabtagene ciloleucel has not been studied in pregnant women. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with axicabtagene ciloleucel, breastfeeding should be discontinued if the mother is treated with axicabtagene ciloleucel. These potential risks may also apply to other agents used in this study * Use of the following: * Therapeutic doses of corticosteroids (defined as \> 20 mg/day prednisone or equivalent) within 7 days before leukapheresis or 72 hours before axicabtagene ciloleucel administration. Physiologic replacement, topical, and inhaled steroids are permitted * Chemotherapy given after leukapheresis to maintain disease control must be stopped \>= 7 days before conditioning chemotherapy * Cytotoxic chemotherapeutic agents that are not considered lymphotoxic (see below) within 1 week before leukapheresis. Oral chemotherapeutic agents, including lenalidomide and ibrutinib, are allowed if at least 3 half-lives have elapsed prior to leukapheresis * Lymphotoxic chemotherapeutic agents (e.g., cyclophosphamide, ifosfamide, bendamustine) within 2 weeks before leukapheresis * Experimental agents received within 4 weeks before leukapheresis unless no response or disease progression is documented while on the experimental therapy and at least 3 half-lives have elapsed before leukapheresis * Immunosuppressive therapies within 4 weeks before leukapheresis and axicabtagene ciloleucel administration (e.g., calcineurin inhibitors, methotrexate, or other chemotherapeutics, mycophenolate, rapamycin, thalidomide, immunosuppressive antibodies such as anti-TNF, anti-IL6, or anti-IL6R) * Donor lymphocyte infusions (DLI) within 6 weeks before axicabtagene ciloleucel administration * Radiation within 1 week before leukapheresis. Subjects must have progressive disease in irradiated lesions or have additional non-irradiated, PET-positive lesions to be eligible. However, palliative radiation to a single lesion, if additional non-irradiated PET-positive lesions are present, is allowed up to 2 weeks before leukapheresis * Prior receipt of CAR T-cell therapy * Participants with signs or symptoms indicative of active CNS involvement are excluded from the protocol, with the following exceptions: The following patients are included in the protocol: * Participants with previously treated CNS involvement by lymphoma or leukemia, who and have no neurologic symptoms and no evidence of active lymphoma or leukemia in the CNS by total spine and brain gadolinium enhanced magnetic resonance imaging (MRI) within 2 weeks before leukapheresis and no neurologic progression prior to axicabtagene ciloleucel (axi-cel) infusion * Participants with active CNS involvement and stable neurologic symptoms for at least 3 weeks before leukapheresis and without neurologic progression prior to axi-cel infusion. These patients should have assessment by a total spine and brain gadolinium enhanced MRI within 5 days before axi-cel infusion to document the extent of CNS disease prior to axi-cel infusion. Cerebrospinal fluid (CSF) sampling for cell count, cytology and cell markers by flow cytometry should also be included within 5 days of axi-cel infusion if previously positive * Inhaled or topical steroids and adrenal replacement doses =\< 10 mg daily prednisone equivalents are permitted in the absence of active autoimmune disease. Participants are permitted to use topical, ocular, intra-articular, intranasal, and inhalational corticosteroids (with minimal systemic absorption). Physiologic replacement doses of systemic corticosteroids are permitted, including if \>= 10 mg/day prednisone equivalents. A brief course of corticosteroids for prophylaxis (e.g., contrast dye allergy) or for treatment of non-autoimmune conditions (e.g., delayed-type hypersensitivity reaction caused by contact allergen) is permitted. Use of anabolic steroids is permitted * The participant has not recovered to baseline or CTCAE =\< grade 1 from toxicity due to all prior therapies except =\< grade 2 alopecia, neuropathy, and other non-clinically significant adverse events (AEs), provided that all other eligibility criteria are met * Opportunistic infection within the last 3 months, with the exception of oropharyngeal candidiasis * History of allergic reactions attributed to compounds of similar chemical or biologic composition to axicabtagene ciloleucel or other agents used in study * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, or psychiatric illness with potential to limit compliance with study requirements