The high-pitched whining you hear is almost certainly from the exhaust
fan.

Your furnace has two fans -- a small exhaust fan to make sure there's a
draft up the chimney to exhaust the combustion gases, and a large
blower fan to blow the air throughout the house. The running of the
exhaust fan coincides with when the gas is burning, which is what you
described.

It sounds like you've got a failed bearing in the exhaust fan, and it's
probably eventually going to fail completely. Ours took at least a
year to completely fail after the first time we heard it making bad
noises.

If you've got a service plan on the furnace, you can ask them to
replace the exhaust fan before it dies completely, but they may say no
(they did when my exhaust fan was dying and I asked them to replace
it). In that case, you'll have to pay them to replace it early, or
wait until it fails, in which case you'll have some period of time with
no heat while waiting for them to come replace it (and perhaps a longer
time while waiting for them to order the correct fan for your furnace,
which is also what happened to us).

Obviously, if you don't have a service plan on the furnace, then
you'll have to pay for replacing the fan whether you do it now or when
it fails completely, so you'll probably want to do it now for peace of
mind.

I'm including your whole article below, because your initial posting
had a typo in the Newsgroups line so it didn't actually appear in
alt.home.repair....

N Harrison writes:
My forced air Gas heater makes a high-pitched whining noise but I'm
pretty sure that it's NOT the blower! If anyone has heating a/c
savvy, please read on.

We are in a house in Chandler, AZ, that is about 4 years old. It has
forced air gas heating. I hear that it's been done cheaply: both the
heater and air conditioner are located in the attic (the a/c
compressors are outside, the blower etc. are in the attic). People
say this is a bad design but cheaper for builders, so we're stuck with
it.

We bought the house in during the summer, so we tested the a/c and it
works flawlessly. I also tested briefly the heater enough to notice
that it worked, but I was so excited looking around our potential new
dream home I didn't notice the high pitched squeal it makes.

So around November, when you have to start using the heater in this
part of Arizona, I switch the thermostat over to heat and am appalled
when I hear this high pitch whining noise that appeared to coincide
with the heater. I figured, no problem, it's probably an unlucky
small vent in a duct that's making this noise, and I'll just find it
and go over it with duct tape. But then I notice that it doesn't
coincide with when the blower runs.

Here's its behavior: when the heater lights up to warm the air BEFORE
the blower starts, the whine begins. It continues while the blower is
blowing, but then about 2 minutes before the blower stops it stops. I
assume this is the same time the fire goes out and it stops actively
heating while the blower continues to blow over the still hot metal
parts.

Does anyone have any idea what is doing this? I could swear it seems
to be coming not from the heater itself but from some ductwork about
20 feet away in the attic (inaccessible to me), and this makes no
sense to me because as I said, it doesn't start and stop with the
blower, so I don't think it's some kind of air movement noise. (I
could be mistaken. In the small attic trying to locate the source of
the whine is like trying to find a cricket--I can't really tell where
its coming from.)

Any ideas?