Does using Devil Fruit powers drain the user's stamina?
It depends on which Devil Fruit. Some fruits drain their user's stamina when its powers are being used, while some don't.

Example: Luffy's Devil Fruit permantly turns his body into rubber. The fruit's power is active at all times, but Luffy doesn't need to use any stamina to remain in his rubbery state. Luffy's stamina doesn't drain simply by existing.

On the flipside, Law's fruit, the Ope Ope no Mi, does drain his stamina when he uses it. He confirms this himself.

So the conclusion is, it depends on the fruit.