Not to convey the same idea. By saying sore there is the idea of walking a lot or being on one's feet for a long period of time. Painful wouldn't be used because it isn't the right agreement, it would be pained feet and that wouldn't work either. Hurt implies injury. You could say tired or aching if you need something other than sore.
One cannot say, "My feet pain." One would say, "My feet are in pain."
Pain implies cause/effect action. Painful would be used to describe the action or incident that caused the feet to be in their condition--not the feet's current condition. So if one cannot say the action in present tense, one cannot say it in past tense with pained.