It could, but I'd start by checking the timing. If it's off the car will overheat, and that's easy to adjust. Heater cores are a grade-A pain to replace because you have to take apart the dash to get to them.
If the Chevy Celebrity radiator is smaller that the Saturn radiator it might well overheat from not enough cooling capacity. If the Chevy Celebrity radiator is the same size or larger than the Saturn radiator I would think it would not over heat.
Yes it can cause it to overheat but first check your thermostat. That would be the 1st cheapist thing to do before you did anything else. I got an 85 regal(overheating) and the thermostat or the heater core is the problem im getting them both replaced tomorrow and will let know of how it runs.