That depends on what make and model of vehicle, some vehicles will overheat if there is no thermostat, others run too cold causing plug fouling high fuel consumption etc.
If you're dealing with a later model, computer controlled, vehicle the engine must reach operating temperature, usually 195°F, within a certain amount of time or the check engine light will be triggered. HTH. -Jim
Pretty much like above todays vehicles need the thermostat say u got one thats for 195 motor will heat up to that point and let coolant flow throughly so u can use heat etc and not having one is just gonna make it overheat. When a vehicle starts up from being cold the thermostat is stuck closed until that 195 degreesis acheived keeps motor cool without overheating. So short answer keep thermostat in or replace if necessary