The thermostat regulates the RANGE of temp that your engine runs in by restricting the flow of cooland throught the sysytem. If your engine is too cold the thermostat (say a 160) will close and allow the temp to build up. When the tempurature rises to 161 degrees the thermostat will open and try to keep the tempurature at 160 degrees by allowing coolant to pass through the rad. Without a thermostat the size of the radiator becomes the main determinming factor in the temurature that your engine runs at. If it is a large enough rad your engine may never reach proper operating tempurature.
This is not good. Generally the cooler an engine runs the more wear it will experiance. I have heard that the rate of wear is somthing like 1000% of normal during the first few minutes of warmup.
The rad in your vehical is designed to keep your engine cool enough to not destroy your oil, or have a multitude of other problems which may occur in extreamly hot conditions which may be compounded by things like headers.
Sooo the odds are without that $15 thermostat that takes what 20 minutes to install? Your engine is going to run too cold, wear out prematurely, and burn a lot of oil.
Guys that race may get away without running a thermostat because they operate in a far narrower band of conditions than a street engine ( especially in northern climates) because they can tailor their cooling systems to their operating situation.
So can you run without one? Yes. Would I? Not a chance.