Gene Therapy-Treated Stem Cells in Treating Patients Undergoing Stem Cell Transplant for Intermediate-Grade or High-Grade AIDS-Related Lymphoma

Inclusion Criteria: * HIV seropositive at or before the time of lymphoma diagnosis * Anti-HIV chemotherapy; subjects must be on a multi-drug regimen (excluding azidothymidine) and have an HIV viral load \< 50,000 copies/ml by reverse transcriptase-polymerase chain reaction (RT-PCR) at the time of study enrollment * Subjects must agree to have their anti-HIV regimen temporarily stopped, and then all subjects will stop antiretroviral therapy (ART) for approximately 7 days at the time they start filgrastim (G-CSF) post-chemotherapy for peripheral blood progenitor cell (PBPC) mobilization and until the mobilization is complete; in addition, if/when the CD4 counts return to a level of 450/mm\^3 with undetectable HIV levels in blood, the subjects will undergo an analytic treatment interruption for an indefinite period not to exceed 6 months * Karnofsky performance status \>= 70% * Biopsy proven intermediate grade or high-grade non-Hodgkin's lymphoma, including plasmablastic lymphoma, primary effusion lymphoma, or biopsy-proven Hodgkin's lymphoma (entities as defined in the World Health Organization \[WHO\] classification); tissue histology will be reviewed at the City of Hope; patients with prior marrow involvement must demonstrate =\< 10% involvement pre-stem cell collection * No psychosocial conditions that would hinder study compliance and follow-up * Pretreatment serum glutamic oxaloacetic transaminase (SGOT) and serum glutamic pyruvic transaminase (SGPT) =\< 2.5 x institution…