In a Poisson model, do we need to check for multicollinearity between the offset variable and other covariates? For example, if I offset by population size, would it be problematic to include covariates that are correlated with population?
No. In a count model, the offset/exposure variable is not a covariate. Rather than being estimated as a coefficient, its value is constrained to equal 1. It allows you to model rates instead of counts. I would review this post for more information.
Incorporating additional predictors that may be correlated with population size should not concern you if you're using population as your offset.