The thermostat holds the coolant in the engine so that it can heat-up more quickly. Then when the coolent heats up and expands it blows open the t-stat and lets the coolent flow freely. You can remove the thermostat and then you don't have to worry about the valve getting stuck and the engine overheating. However if you just replace it ever time you flush the radiator you won't have any issues. Barefoot, the thermostat does the opposite of what you are getting at, it holds coolent in the engine so that it gets hotter, not in the radiator. Netjaws is right throwing a belt will cause the engine to overheat, but it will also kill your powerstearing, and alternator, so most of the dash will light up. Without a fan belt you can really only travel about 5-10 minutes without overheating.
DelxE, just keep track of when you last replaced the t-stat and you should be fine, on the D2 I beleive it's plastic so you don't even have to worry about it rusting.