Sulfur‐based stratospheric aerosol intervention (SAI) can cool the climate, but also heats the tropical lower stratosphere if done with injections at low latitudes. We explore the role of this heating in the climate response to SAI, by using mechanistic experiments that remove the effects of longwave absorption of sulfate aerosols above the tropopause. If longwave absorption by stratospheric aerosols is disabled, the heating of the tropical tropopause and most of the related side effects are strongly alleviated and the cooling per Tg‐S injected is 40% bigger. Such side‐effects include the poleward expansion of eddy‐driven jets, acceleration of the stratospheric residual circulation, and delay of Antarctic ozone recovery. Our results add to other recent findings on SAI side effects and demonstrate that SAI scenarios with low‐latitude injections of absorptive materials may result in atmospheric effects and regional climate changes that are comparable to those produced by the CO2 warming signal.