I think you can probably find a way to move your thermostat. I would be surprised if there were not a battery-operated extension device available.
It’s pretty much like a wired telephone. It is recommended to turn off the furnace when messing with the wires, though. The voltage is low, but why confuse the poor furnace?
Ah, good idea! What we’d like it to get the current one moved to another room downstairs, maybe the schoolroom (which would be the dining room if we used the house as planned by the builder). That would get it well away from the stove and the fire.
Then we’d like a separate thermostat upstairs, and the ability to control the ac/furnace from one or the other, as we choose.