When a professor takes an affiliate appointment in another department, do they get an increase in salary?
I have never heard of a joint appointment in a second department being tied to a salary increase. It’s possible that in some rare situations such an increase will be negotiated as part of the deal for a professor to take on a joint appointment, but generically, the answer to the title question is “no”.

On the other hand, the answer to “or is it simply more responsibility for no incremental salary, kind of like service work?” is also “no”. Being a joint appointee does not involve “more responsibility”, rather it’s more accurate to describe it as a shift in responsibilities. Professors without a joint appointment already have a full time job, so they cannot increase their responsibilities (at least in a formal sense, and to a large extent also in a practical sense). If a professor takes on a joint appointment in another department, it’s likely because the professor simply likes the idea of being involved in the other department because their interests are aligned with that move, in the sense that the professor wants to have collaborations with other people in that department, wants to advise students in that department, wants to teach courses offered by that department, or similar things. But they had a full time job before the change in appointments, and they have a full time job after the change in appointments, so the total amount of work they are expected to do overall does not change.