The overheating might have damaged the battery (or in itself may have been the death throes of the battery).
Before you bring it to a repair shop, try plugging it into a wall outlet and let it charge undisturbed for at least 6 hours. Then press and hold Power for about 15 seconds to see if anything happens. If nothing happens, then press and hold Power and Vol Down for about 15 seconds to force a shutdown (in case the phone is actually frozen on a blank screen) -- then try to power on normally after that.