So the feed stays in the coop?
My feed stays in the coop but I also put out feed bowl in the run in the morning (to provide fermented feed, and to spread the birds out and keep them from crowding around a single food source).
Ridgerunner's post explains it well - do what works for your set up.
But if you do keep food in the run instead of the coop, it absolutely should be picked up or sealed up at night to deter rodents and wild birds. Even with a predator proof set up there's no reason to try and tempt pests to break in.