Can we say Tehillim on all fast days or are there some fast days we can’t say Tehillim?
We are allowed to say Tehillim on fast days. In fact, a fast day is really meant for us to repent and improve ourselves, and saying Tehillim is great thing to do on a fast day. The only exception to this is Tisha B’Av, when we generally don’t say Tehillim unless it is for a sick person, etc.
Magen Avrohom 454-6.