Failure of your thermostat to get power is just one reason your furnace or central air system won't turn on, but it's one of the easiest to check, so it's a good place to start in the process of diagnosing the problem. It's an electric component that functions like a switch, and the easiest way to test whether it's getting power is to bypass it and see if the furnace switches on. If you have a multimeter, you can also use that to test the thermostat for power, and this test is even easier to conduct than the bypass test.
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The thermostat operates at low voltage (24 volts), so if you get a shock, it won't be a bad one. Nevertheless, no shock is better than a mild one, so play it safe and turn off the power whenever you're handling the wires.
Conduct a Simple Pretest
You might not even have to remove the thermostat cover to find out if the unit has power. One simple way to check is to turn up the thermostat as far as it will go and wait a minute to see if the furnace turns on. If the thermostat controls an air conditioning system, turn it down as far as it will go. While you're at it, toggle the "fan" switch from "auto" to the "on" position, and if the thermostat has power, you should hear the blower start.
The fact that the LED on the thermostat screen is lit doesn't mean it's receiving power from the transformer. The LED and the programming functions are all powered by a battery.
How to Conduct a Bypass Test
If none of the pretests are conclusive, the next step is to remove the thermostat cover to expose the wires. If you don't see wires, you're looking at the mounting plate for the temperature sensor. This is usually held on by two or more screws that you can remove with a Phillips screwdriver. When you get to the base plate, you'll see a number of colored wires attached to terminals designated by letters. The ones in which you're interested for this test are the wire attached to the R terminal, which is usually red, and the wire attached to the G terminal, which should be green.
Turn off the power to the furnace or air system if you haven't already, disconnect these two wires, twist them together and turn the power back on. The G terminal controls the blower motor, so if the thermostat is receiving power, the fan will start. You can also conduct this test with the red power wire and the (usually white) wire connected to the W terminal. It controls the heat and will also make the blower come on if the thermostat has power.
Test the Thermostat With a Multimeter
If you happen to have a multimeter, you can test the thermostat without turning off the power or disconnecting any wires. Turn the meter dial to measure 24 volts AC (usually denoted by "VAC" or "V" with a squiggly line over it). Touch one of the meter probes to the R terminal (to which the red wire is attached) and the other probe to the G, W or Y terminal and note the reading. If the thermostat has power, the reading should be between 22 and 26 volts. If the reading is 0, the thermostat isn't getting power.