The fan may not coming on. The water pump may getting bad. Is water leaking out the weep hole. There must be a hole in the housing. or the water level is low. The temp is actually HOT, and when stopped the temp shows cool because there is no water on the sender.
When you are driving the temp comes up because the water is flowing and passing over the sender. The heater is just a small radiator, in parallel with the main radiator. You have a problem elsewhere, and the heater core when working removed enough heat to keep things under control. Once the heater core no longer removed heat, THEN the system was overloaded. It means you have a problem with other parts of the main cooling system which you did not see while the heat removal by the heater masked the problem, and then when it no longer removed heat, resulted the overheating of the engine. A heater by itself has NO effect on the operating of the engine other than to take some heat and shunt it into the passenger compartment for your comfort. Think of the heater core as a small radiator because that's what it is. The hot coolant flows through the inside of it while the fan blows air through it and into the passenger compartment. The only way it could contribute to overheating would be if it was leaking causing a loss of coolant. This should be easy to spot since there's going to be coolant either under the car or there will be coolant on the floorboards when the engine is running and the heater is turned on. The usual tip-off to a bad heater core is the smell, the slightly sweet odor of hot coolant and sometimes the lack of heat when its turned on.

Your problem could be a bad water pump, or a clog somewhere in the coolant system. Have you tried flushing the system with a cleaner? This will help. Thanks please keep updated.please do rate the solution positively .thank you for using fixya