Place chicken breasts in the middle of a rimmed sheet pan. Add green beans on one side and potatoes on the other. Pour melted butter over everything and sprinkle with Italian dressing mix. Cover with aluminum foil.
Bake in the preheated oven until chicken is no longer pink in the center and the juices run clear, about 60 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).