My Heil Heat pump won't shut off without hitting the breaker. I have already replaced my thermostat. Any Ideas?

There is a contactor in your heat pump that the 24 volts operates, it could be that the points on this contactor are stuck together and therefore will not cut off the power to the heat pump.

Also I want to ask you what you mean by "hitting the breaker", I am assuming that you mean that you are turning it off.

What part will not turn off? The compressor,the fan, or both

If it's the heatpump stuck on it may be just a stuck contactor, but if the heat pump is running when it shouldn't and your turning the breaker going to the blower section off and it quits you have a 24v problem at the thermostat or thermostat wire going to the blower section or heatpump.