The system works opposite to normal - closed circuit means the system is good, open means the system is bad. meaning that please check the wire that goes to the heat switch, if its disconnected or burnt or the sensor is bad then the light will not go out.
For your overheating problem, check the water pump and cooling system for integrity, you should also closely inspect the main crank pulley damper for wear and damage, if its worn out and loose your engine will heat up as the water pump rpms are slowed down.
p.s. I am believing you have a 2E engine in your car.