Car Overheats unless gas pedal pushed down

I should have mentioned the car type:

1991 Lexus LS400
235,000 Miles or so
Don't know engine size - I know they didn't vary a lot in 1991. :)

I am having the weirdest issue.

My heater doesn't work unless the gas pedal is pressed in to a certain point. My car engine's temp. Will eventually get really hot and sometimes overheat if its not pressed it as well. I'm not "so" concerned with the heater as I am with the engine temp. The water pump has already been replaced as well as other general water pump/collant related maintenance work. (Sorry for the generalization - not much of a car person) When I took it in to have the pump replaced I spent a load of money just to not have the problem resloved. Does anyone have any ideas in regard to this problem?

The overheating could be a very overlooked issue on these cars. The fan clutch, which on the V-8 models are reverse rotation and if you dont have the right one or a weak / worn out one it will simply overheat.
Try putting a box fan against the grille and let it idle. Watchn the temp and if it stays normal, then, walla!

Have you put any stop leak in your radiator? To me it sounds like something clogged up maybe your radiator and water can't flow properly and when you give it gas it's putting more pressure on it to make it flow.