So I read that storing raw potatoes in the fridge causes the starch to turn to sugar (which I don't want). So I store them in the pantry shelf. But what about cooked potatoes? Will storing them in the fridge cause the starch to turn into sugar?
The advice for storing raw potatoes is because they are still "alive" in a sense.
By storing them in the cold, you are telling them it is winter and they should be converting their stored starch into sugar so they can grow in the spring.
Once you've cooked them, that's no longer a problem - you've killed them, so the conversion isn't going to happen any more. General food safety rules would then apply.