Beat a dead horse: The point being, a dead horse is dead and can't feel anything. Continuing would be gruesome but ultimately also very pointless.
Feed a fed horse: But then you'll get a fat horse. Continuing maybe not so good for the health of the horse, but if you want to make good friends with said horse? This would be one way to do it. So... in a perverted sense, there's a point to the action.
Beat a dead horse ≠ Feed a fed horse.