If you turn the power off to the furnace, does outdoor unit continue to run?
Has this always been an issue or just after a particular project?
and is this ac or heat pump?
You are correct.
I guess to move forward I would need furnace info and then I can look at proper schematics
And to verify is the condenser fan AND compressor running non stop?
What color wires do you have on what terminals in heat pump? I refer to the thermostat wires on R,Y,O,T.......
The 3 wire cable that is attached to R via the white...we should have only 2 wires hot from R...one to thermostat and one to R in outdoor unit.
Is the 3 wire cable only using 2 of the conductors?
If no wires on O then they have not landed them on air handler strip...and must have twisted O from thermostat to O to heat pump. O = reverse valve and is necessary for heat function.
2 wires on Y..one to outdoors and one to thermostat?
The Y wire that goes outside is the only wire that starts the compressor...the R is 24v constant, Y = compressor, O = reverse valve.
Somewhere you have a short of 24v crossing from R to Y. This will take elimination...but the yellow wire coming from outdoor must be traced to see where the 24v is being connected to it.
My bet would normally be thermostat wiring but you said it continues to run with thermostat off the wall.
This means the yellow wire is picking up 24 from the R teminal wires somewhere.
I wish I could help more....but the best I can do is recommend tracing yellow wire and making sure it only goes to Y on thermostat and Y wire direct to Y in heat pump
Then your breech is at the r wire going outside and the yellow wire going outside.
But also, you said when thermostat disconnected the compressor didnt run...and the fact you removed Y and it still runs doesnot support the thermostat being off with no opertion.
With Y unhooked and unit still running can only be 1 of 2 things...
Short from r to outdoor unit to Y in outdoor unit...and then a mechanically stuck contactor.
But you said the unit shut off when power turned off to air handler.
With that info and the removing Y...your issue is from the indoor unit to the outdoor unit and involves power getting from the R wire to the Y terminal.