A car can run just fine without having a thermostat in the cars engine coolant system and it will not overheat.
However the heater in the car will not work properly when you do not have a thermostat in the vehicle because the coolant just constantly flows through the engine.
That keeps the heater core from getting enough coolant that is hot to allow the heater to work.
A cars engine will actually cool off better without a thermostat but you really do want to have a thermostat in the car if you want to keep your heater working.
If the thermostat is stuck closed then yes the cars engine would eventually overheat because the car is not getting enough coolant flowing through the engine.
But having the thermostat stuck open or not having a thermostat at all will do no harm to the engine.