There are many signs, but some of the more benign are simply that the furnace might not come on as soon in cold weather conditions, or it might go off when it's supposed to be on. Other problems can include a temperature differential between rooms or improper cycling with changes in outside temperatures. The upside is most thermostat issues can be fixed by someone who knows how to swap out a wiring part and fix a blown fuse. In rare cases, though, an electronic control may need to be replaced entirely so it may require professional assistance for repair.