The primary human immunodeficiency virus (HIV) reservoir is composed of resting memory CD4+ T cells, which often express the immune checkpoint receptors programmed cell death protein 1 (PD-1) and cytotoxic T lymphocyte-associated protein 4 (CTLA-4), which limit T cell activation via synergistic mechanisms. Using simian immunodeficiency virus (SIV)-infected, long-term antiretroviral therapy (ART)-treated rhesus macaques, we demonstrate that PD-1, CTLA-4 and dual CTLA-4/PD-1 immune checkpoint blockade using monoclonal antibodies is well tolerated, with evidence of bioactivity in blood and lymph nodes. Dual blockade was remarkably more effective than PD-1 blockade alone in enhancing T cell cycling and differentiation, expanding effector-memory T cells and inducing robust viral reactivation in plasma and peripheral blood mononuclear cells. In lymph nodes, dual CTLA-4/PD-1 blockade, but not PD-1 alone, decreased the total and intact SIV-DNA in CD4+ T cells, and SIV-DNA and SIV-RNA in B cell follicles, a major site of viral persistence during ART. None of the tested interventions enhanced SIV-specific CD8+ T cell responses during ART or viral control after ART interruption. Thus, despite CTLA-4/PD-1 blockade inducing robust latency reversal and reducing total levels of integrated virus, the degree of reservoir clearance was still insufficient to achieve viral control. These results suggest that immune checkpoint blockade regimens targeting PD-1 and/or CTLA-4, if performed in people living with HIV with sustained aviremia, are unlikely to induce HIV remission in the absence of additional interventions.