Do triggers from items have to be activated if the trigger occurs?
For instance, I've got an Owlbear Claw on a weapon, and I critically succeed on something I don't want to have it used on.
Not as far as I can tell
No, you are not required to use the free action to activate a talisman. Thus would only change if the talisman itself had a more specific rule, requiring it to be used at first opportunity.