You can drive a car with a bad thermostat as long as the thermostat is only staying open and not stuck closed.
If the bad thermostat in your engine is stuck open and not closing then it's okay to drive your vehicle with the bad thermostat being stuck open.
However the engine will not be able to get warmed up properly and the heater will usually not work with a bad thermostat.
If your engines thermostat is stuck closed then no you cannot drive the car with the stuck closed thermostat because the engine would overheat.
The coolant needs to flow through the thermostat and into the engine and back to cool the engine.
The engines thermostat works by responding to the changes in the temperature of the engine coolant and when the engine and coolant is cold the thermostat is in the closed position.
The thermostat then stays closed until the engine warms up the coolant so you may see your temperature gauge go up when first starting the vehicle which is normal.
If the thermostat is good and opens then the temperature gauge should go back down and if it does not then your thermostat could be stuck closed.
The thermostat opens once the coolant is hot enough to allow the coolant to flow through the radiator.
A new vehicles thermostat is only about $5.00 or $10.00 and they are pretty easy to replace but you need to drain some of the coolant and remove the thermostat housing and replace it with a new gasket.
A shop could replace the thermostat for around $50.00 to $100.00 in parts and labor depending on how hard it is to get to the thermostat.