I know what happens electrically with my current old oil furnace when
the oil doesn't ignite. A relay is tripped that has to be reset
mechanically with a red button.

But someday soon I may need a new furnace and I don't know how they
work.

Here is the reason: I'm getting a new burglar alarm (mine went up in
smoke, literally) and my friend told me they had a built-in temp
sensor, to notify them if the house temp got too low.

It turns out this model requires an add-on switch, and while trying to
decide where to put it and how to wire it, it occurred to me: Why
wait until the house is getting too cold. Then if I'm away, my
friends only have a day or two to get there and fix the furnace and
warm the place up. Why not also notify the monitoring company there
is a problem when the furnace first tries to ignite and can't (no
fuel, nozzle clogged, furnace broken, etc.)

My current furnace has a mechanical relay, a latching relay I
presume????

and if it is not double pole or double throw, I could buy a latching
relay that is, and use it instead and connect the unused pole to
notify the alarm that the furnace needs resetting. I would give that
its own zone, so that I would notice it even when I was home, when
setting the alarm.

But I'm guessing everything is transistorized in a new furnace I buy.
I don't have trouble working with IC's and circuit boards, but that
doesn't mean there will be a place to connect. Is there any way to do
the same thing with the new furnace? Is there a brand of furnace that
would make this easier than other brands?