Doug Smith writes:

How usefull are the 10-year extended warranties for new furnaces, such as
for Trane, Carrier, Lennox, etc. My gut instinct is that I don't want to
spend $4500 on a new furnace that is likely to break down in under 10
years...

It may help if you realize that that warranty is nothing more than
insurance.

My own rule is never insure what I can afford to loose.

I can't afford to loose my home so I insure it.
I can afford to replace the furnace so I would never insure it.
Your average loss on a furnace is a few hundred dollars, not the
total cost.