Is it a good or bad idea to include a password's salt in the calculation of a MAC?
It seems unwise to give an attacker enough information to run an off-line attack against a password even if you're using a good PBKDF. PAKE would probably be better.

Assuming you have a strong key your question still holds. I interpret it as:

Given a key $k_1$ and a salt $r$ you derive $k_2 = KDF(k_1|r)$. You then send $m, r$ and $MAC_{k_2}(m|r)$.
Is it safe to do this given that a change to $r$ will change both $k_2$ and $m|r$ or is there a chance that $MAC_{k_2'}(m|r') = MAC_{k_2}(m|r)$?

And my answer is, there is no chance of that happening because it would constitute a forgery. Given a good KDF $k_2'$ will be indistinguishable from random so a valid MAC for anything under that key including $MAC_{k_2'}(m|r')$ would be a forgery.