How To Test a Thermostat
Before you assume your furnace or A/C is malfunctioning, try these simple steps to make sure it isn't a problem with your thermostat. Thermostats are simply a switch; they turn the furnace on and off. The only big difference from a light switch is that they have a mechanism that measures the temperature and when it reaches a predetermined point, it activates.
If you turned on the thermostat and the heater (or A/C) didn't come on, it may be a problem with the thermostat and not the furnace. First things first, make sure that all the settings on the thermostat are as they should be. Make sure the "ON" switch is on, make sure it is set to "HEAT" or "AC" as appropriate, make sure there are no programming overrides. I've turned up the heat and had nothing happen because I failed to notice that someone turned it to "AC".
If your thermostat is battery operated, try installing a fresh battery. Even if the display is lit, a weak battery may be a culprit.
If the thermostat is properly set and you have a fresh battery, then the next easiest thing to check is best done with two people. One person operates the thermostat (with the thermostat set to heat) turning it slowly from a low temperature to a high temperature. The second person is stationed by the furnace (even if the problem is with the AC, test the furnace). When the thermostat is turned on, it usually makes some sort of clicking sound which indicates a signal is being sent. At the furnace, when a signal is received, it should make a sound. If it makes a sound, any sound, it is almost certainly getting the signal and the problem is with the furnace. If it is dead silent, then the problem is most likely in the thermostat or connective wiring. If you don't have a helper, try using a walkie talkie with the talk button taped to the "talk" position or a baby monitor or even use a cell phone to call your home phone. Leave the transmitter near the furnace and listen from the receiver for a sound.
The next test is a little more difficult, but not too tough. Your thermostat is a switch, right? Well, let's bypass that switch and see what happens if we connect the wires together. That will do the same thing as turning on the thermostat. In order to conduct this test you will have to access the sub-base. Most thermostats have an outer shell and beneath it is the sub-base. Some models simply pop off when pulled straight off or pull from the top or bottom first then off. Other types may require a twist, depress a tab or may be held in place with one or more screws.
Before proceeding, turn off the power to the furnace. The thermostat control is low-voltage and poses little risk, but best practice is to turn off the power to avoid injury or damage to equipment. The sub-base is screwed to the wall. You will probably see two or four wires coming from a hole in the wall and connected to screw terminals on the sub-base. If you have four wires, this indicates wiring for both a furnace and central air conditioning. If you see just two wires, those are for the furnace.
Before you disconnect anything write down which wire connects where or take a photo. Sure, you think you will remember, but 15 minutes from now you will be thinking "is it the red wire or the green wire." Even more important, if you disconnect any wires, secure them so them don't fall back through the hole in the wall. Use a small spring clamp or wind the wires a few times around a pencil to prevent the wires from dropping through the hole and into the wall.
To test the heater you will need to connect the "R" (or "Rh") wire and the "W" wire. This bridges the two wires and has the same effect as turning on a switch. Turn the power back on and listen again to the furnace for any sounds. If the furnace starts up, then the problem is almost certainly the thermostat and you should replace it. Do not leave the furnace running like this for an extended period of time, it will not shut off and could overheat or cause a fire. If the furnace is completely silent, make sure the furnace has power. Check the circuit breaker and use a tester to make certain there is power at the furnace. If it has power but does not start, then the problem could be a break in the wire running from the thermostat to the furnace or with the furnace itself.