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 antivi