Before you do anything, get the inspector to clarify exactly what they want, in writing, so that there is no confusion, and "I didn't mean that, you have to do it over" moments.

If they want to prevent smoke from being spread throughout the house, using a relay to interrupt the thermostat won't do it. If the furnace is on and producing heat, or if the fan is switched mechanicaly on to continuously supply air circulation, the fan will continue to operate at least until the furnace cools down, as either the mechanical switch or the high limit switch will keep power to the fan motor.

It would seem that the smoke detector relay would have to operate a 120 volt relay to cut power to the whole furnace, so that everything, including the fan is shut down when detecting smoke.