2002 Chrysler Concord

Check your coolant levels. a heater core is always "on" in that there is always engine coolant flowing through it. Turning on the blower motor for cabin heat should have no effect on it, outside of providing a miniscule amount of cooling as it does act like a mini-radiator. If you're borderline overheating, the cause is likely to be low coolant, restricted coolant flow, bad thermostat, bad water pump, bad engine coolant sensor. Start with the basics, good luck.

Not all heater cores have coolant circulating through them all the time but Ken is right - Check the Coolant level.

You're right FoR, I see the Concorde does indeed have a control valve for the heater core - so back to the OP's problem, that valve opening could be enough to lower the coolant level in the system to cause the overheating.