Do I turn off the switch labeled Furnace or the switch labeled Air Conditioning on my circuit breaker?
To be certain you have cut power to your thermostat, switch both circuit breakers to off. Then, confirm that the power is off by increasing or decreasing the temperature on your thermostat and waiting five minutes. If during that time you either hear or feel your system turning on, you have not successfully cut the power to your thermostat.
Note: Many of today's digital thermostats are battery powered, which makes it difficult to tell if the power is off. Listen to ensure your system is turned off before continuing.