I have a LENNOX heat pump system that is about 3-4 years old (originally installed with house). Yesterday, I set the COOL to 76 degrees and the compressor outside and fan inside would not turn off when it reached "76 degrees". It continued to stay on, even when the temperatures reached 70 degrees on the thermostat.

I had an issue like this before and replaced the Thermostat which fixed the problem, so I replaced the thermostat today and the problem still happens. When I turn the thermostat to off, it will turn both the blower and the compressor outside off.. HOWEVER, sometimes 20 minutes later, the compressor outside will turn on by itself, but not always. This is not true for the fan, the fan will remain off if the theromstat is on the off position.

Any help would be appreciated!

To me it sounds like you have some low voltage wires shorted together at the thermostat or going to the heat pump or fan coil. O to Y or G circuit or R to Y or G. Check your thermostat wires from the thermostat to the fan coil an then to the heat pump for a short.