I sometimes use an offset in a logistic regression model. The use case is where I already have a complex model, which needs to be re-estimated to cover some new data outside the realm of the original data sample (in time, or in cross section), but where, for various reasons, it is practically infeasible to re-estimate the model on the entire, expanded data set. The goal is a new model that gives good predictions on some out-of-sample data, but which gives unchanged predictions on the in-sample data.
So I take the linear predictors from the original model, specify those as an offset, and then introduce additional variables aimed at fitting the new data, in such a way that it wouldn't change the predictions on the original data.
It's admittedly ad hoc, but an awfully useful trick in practice. I have no idea what the "legitimate" use of an offset in logistic regression is, but I'm glad statistical software packages allow for it.