Are you positive you have the correct breakers?
Some may seem to fit but have loose connections to the panel which causes heat to be generated under a load.
Only breakers listed for a particular panel may be used.

Thanks so much for your reply. I took the old breaker to the hardware store with me, and the person at the store said it was the same, although the screws to hold the connections in were bigger, as was the opening where the connections go into the breaker. Everything else looked identical. He had a hard time getting them to snap into place in the panel, though. He said they were a really tight fit.

I noticed when he installed them, he was dabbing some stuff on the connections before he slid them into the new breakers. It was in a bottle that looked almost like an Elmer's Glue bottle. Whatever that is, could that be the source of the smell?

The "guy at a hardware store" is rarely a reliable expert on things electrical. I have heard more drivel spouted by these guys. I usually have to grit my teeth on their opinions (though I had to throttle a guy once when he told a customer to do something patently unsafe).

Heating contractors are frequently also unqualified to do electrical work. I suspect the goo he applied was antioxidation compound which shouldn't have made too much of a difference. Call in a real electrician. No matter whether these guys were right or wrong, you have an electrical issue now.

We are not told what kind of "furnace". We are not told the ampere ratings of the breakers. If it is an electric furnace, the breaker problem is more understandable. If oil or gas, they typically have low amperage requirements.

I would be really concerned at why the original breakers melted. Breakers don't spontaneously melt. That was the symptom of a much bigger issue. The smell might be arcing. Get and electrician and quickly . This could very well be a significant fire hazard.

Pull the new breakers and look at the buss they attach to for damage. Breakers meting says something is very wrong, probably there.

If you have the room in the panel, try moving the breakers to another position on the buss, and see if the smell/heat stops.

Hmm I wonder if the furnace was wired incorrectly? There wasnt a problem before the HVAC tech messed with the furnace wiring?

I should have also asked what is the brand name on the breaker and on the panel. I hope neither is Federal Pacific.