Antibody therapy for HIV-1 infection exerts two broad effects: a drug-like, antiviral effect, which rapidly lowers the viral load, and a vaccinal effect, which may control the viral load long-term by improving the immune response. Here, we elucidate a trade-off between these two effects as they pertain to the humoral response, which may compromise antibody therapy aimed at eliciting long-term HIV-1 remission. We developed a multi-scale computational model that combined within-host viral dynamics and stochastic simulations of the germinal centre (GC) reaction, enabling simultaneous quantification of the antiviral and vaccinal effects of antibody therapy. The model predicted that increasing antibody dosage or antibody–antigen affinity increased immune complex formation and enhanced GC output. Beyond a point, however, a strong antiviral effect reduced antigen levels substantially, extinguishing GCs and limiting the humoral response. We found signatures of this trade-off in clinical studies. Accounting for the trade-off could be important in optimizing antibody therapy for HIV-1 remission.