Thermostat

Not unless you want the car to overheat. No vehicle can run without a thermostat. It would cause the motor to over heat and blow your head gaskets. If you have a thermostat off, you're going to need to get it replaced to make sure the temperature of the motor is regulated. This is an unavoidable fix.

If you physically remove the thermostat from the cooling system, it can keep the engine's computer in "open loop" mode, costing you performance and mileage. The engine won't be able to properly warm up and computer will resort to running too rich. Running too rich can cause a host of other problems, as well as burning up your catalytic converters. Besides, if you can open up a thermostat housing to remove it, then surely you can afford $5 to put a new one in.