I would like to install a second thermostat on the lower floor of my home. My thinking is that when we have guest staying on the lower floor I could turn the upper floor thermostat off and the lower on and the heater would respond to the temperature on the lower level. Would this arrangement work?
This would work if wired in parallel. When either stat calls for heat it would activate the furnace. The only problem I could see if you used smart thermostats that combined may draw more than the control transformer can supply but if simple non powered thermostats I don't see any problems.
The other issue will be that if the temperature sensor is in the basement, the upper part of the house may get too warm by the time the furnace displaces all of the cold ait in the basement. Remember, warm air RISES, so if your return air resister is not in the basement, it will be a long time before there is enough air exchange to make a difference down there.
What I would do (have done) is to add a baseboard heater as a supplemental system, with its own thermostat in the basement, along with a way to keep the basement air from rising out of the basement (i.e. stairwell door).