When a test indicates a leaking thermostat housing gasket and it has to be replaced would you need to drain the cooling system before you can remove the thermostat housing?

No just make sure that the engine is cool. take the radiator cap off MAKE SURE ENGINE IS COOL... then you can take the thermostat housing off and replace the gasket you might have to replace the thermostat as the gasket might be an O ring around the Thermostat and not bought separatly. after your done fill up with water.