Is it loosing coolant? It sounds like a blown head gasket and the exhaust may be entering the coolant system? It could be a few things. Sorry to be the bad news guy..
They sell a kit that detects exhaust gasses in the coolant that might be worth picking one up.
Take a read though this thread.
http://www.toyota-supra.info/forums/mkiii-faq/13-help-my-car-is-overheating.html