We explore the effects of continuous number density measurement on atoms in an optical lattice. By integrating a master equation for quantum observables, we calculate how single-particle correlations decay. We consider weakly and strongly interacting bosons and noninteracting fermions. Even in the Mott regime, such measurements destroy correlations and increase the average energy, as long as some hopping is allowed. We explore the role of spatial resolution and find that the heating rate is proportional to the amount of information gained from such measurements. © 2014 American Physical Society.