Could overheating be caused by a bad heater core in a Jeep?

I wouldn't think so, unless it let all the water leak out. Typically in the summer time you are running the vehicle with the AC on and no water is circulating through the heater core, and your vehicle doesn't overheat.

If your heater blows out hot air then it is good. Try replacing your thermostat. If not a clogged heater core could demonstrate particles floating around your cooling system that may ultimately foul your radiator.

I'm going to add to this answer. Without a doubt a clogged heater core can cause overheating on some engines. GM's 2.5L Tech 4 engines are just one of many that will over heat if a heater core is restricted. The way the cooling system is designed on this engine will not cool properly at hard acceleration or high idle with a clogged heater core. In this engine the heater core is fed by coolant from the lower thermostat housing first going through the intake manifold then to the core and then returned back to the radiator via independent heater hoses. If a clog in any part of this would happen it will cause the coolant to stall at the thermostat housing. Usually you will see normal scan tool temperature readings at high throttle, but the temp gauge will usually climb into the red at the same time and quickly return to normal once throttle has been taken back down to idle. You will also notice signs of high cylinder head temps on it's painted surfaces or hard burnt spark plug wires can also be a sign if problem has gone on too long. Blown head gaskets and cracked heads can also result from this very quickly.