Always. The "restaurants don't refrigerate" claim is dumb once you realize they go through a bottle in a few hours or days. You don't do that with your average hot sauce. Hot suace loses heat and flavor fast once opened.
Also, anyone claiming that they don't put their homemade salsa in the fridge is doing it all wrong. Sure you don't refrigerate fresh or canned tomatoes but you certainly should after pulsing it through your Cusine-Art and adding all sorts of other ingredients. Just because it's technically safe to eat doesn't mean it'll taste as good.