AC unit turns on and off when it's in heater mode. Doesn't do that when it's on air condition mode. Any suggestions?

Check for voltage, if it has voltage is there an amp draw? If not its bad.You can ohm the motor leads to see if its open.

Is it a window unit or a central air unit? A dirty filter or dirty coils will cause this to happen on both types because it is overheating in heat mode from lack of airflow through the indoor coil. This will cause the compressor to fail unless you clean the coils with coil cleaner (Always use alkaline type coil cleaner with safety equipment such as gloves and face shield. Never use acid type; it has hydrofluoric acid in it.).