What happens if the batteries are proven to be good?
some brands of thermostat are known to be unreliable
unmount the thermostat from the wall and then connect the two wires going to the furnace or use a jumper
if the furnace activates then the thermostat has failed
the older mercury thermostats were far more reliable than the newer non-mercury ones
a thermostat is wired to 24 volts DC and the batteries are solely for the benefit of maintaining your preset programs.
the 24 volts only runs thru the thermostat to activate the relay in the furnace