While I'm sure you can figure out a way to do this, I doubt it's worth
all the trouble, such as trying to decypher and wire into the
electronicc ontrols of a brand new furnace. As noted, the burglar
alarm has the ability to monitor temp. As long as the temp to set off
the alarm can be set to just below the temp you keep the house at, say
3 to 5 degrees, I don;t see where the furnace alarm is going to give
much additional benefit. If the alarm doesn't allow a temp you like,
but has external inputs, I'd just hook up a termostat to it.

I see 2 scenarios. One, it's winter and 10 degrees outside and the
furnace fails to start. The only thing you gain with the furnace alarm
is the time it takes for the house to drop 3 degrees, which might be an
hour or two at most, which hardly seems worth the trouble, as you
still have quite a bit of time before anything is going to freeze.
Or other scenario is it's Fall and 40 outside, in which case it may
take 6 hours to drop and trigger the alarm. But, in that case, you
have days to fix the furnace.