I have a White-Rogers, model #1F80-51, thermostat. The thermostat chirps a few times every minute despite the fact that I have changed the batteries out, checking them with a multimeter to make sure the new batteries are of the proper voltage. What should I do next?

You may have to go to White-Rogers, either for any instructions you have for it, or I found their website very helpful. It is possible the batteries are only back up and it is trying to tell you it needs a connection to the common side of the transformer at the furnace. Some digital thermostats need often a blue wire connecting a B or C terminal at the furnace to the B or C at the thermostat. Also usually there should be a jumper on the thermostat between RH and RC. Even if the wires are there, the conductor could be broken under the insulation.