A vehicles thermostat is there to help the engine reach operating temperature when you start it.
The thermostat in a vehicles engine starts at a closed position which allows the coolant in the engine to heat up and allows the engine to reach operating temperature.
The thermostat in the engine also works with your vehicles heater to allow the coolant to get hot enough in the engine to allow hot coolant to flow through the heater core which then allows the hot air to be blown into your vehicle.
If you drive without a thermostat in your vehicle or if the vehicles thermostat is stuck open then your vehicles engine will not reach operating temperature.
Your heater may also just blow cold air as a result of a stuck open thermostat or no thermostat at all.
Thermostats for engines are pretty cheap and easy to replace in most vehicles so I would not recommend driving without one.
If your coolant thermostat is stuck closed then the engine can overheat because it's not allowing coolant to flow freely throughout the engine.