If Heater core is bad on a 1984 Mazda GLC will car overheat?

Not necessarily. It depends on what you mean by a bad heater core. The core will either be leaking or stopped up. If it is leaking, then it can cause the engine to overheat and the only fix is to replace it. A clogged or plugged heater core can sometimes be fixed by flushing it. You can also temporarily bypass the heater core until you can get it fixed. Just connect the inlet & outlet heater hoses together.