Recent experiments with optically levitated particles have shown great promise for high-precision sensing of accelerations and gravitational fields as well as exploring mesoscopic physics. One barrier that often stands in the way of improved acceleration sensitivity or quantum state coherence time is high particle temperatures due to the absorption of the light from the trapping laser. In optically levitated acceleration sensing architectures, one limitation on the precision of such sensors is often the upper limit on the size of the particle that can be trapped: larger particles require more laser power to levitate, but too much absorption of the trapping light can overheat and vaporize the particles. We present a detailed analysis of a levitated optomechanical accelerometer to understand what combinations of acceleration sensitivities and maximum-tolerated accelerations can be reasonably achieved, and we analyze the extent to which anti-Stokes optical refrigeration may solve the problem of overheating particles. We also analyze the effect of blackbody radiation (BBR) pressure shot noise on a force and acceleration sensor concept involving free-falling particles that are released and recaptured by an optical trap. We find that, although optical refrigeration is likely insufficient to solve the problem of large particles vaporizing in high-power traps, it would help mitigate BBR pressure shot noise in future accelerometers based on free-falling particles.