The 20-quart stainless brewpot I have doesn't have the tri-clad bottom. It works well enough on my apartment stove, but will it be a problem if/when I upgrade to a propane burner? The pot isn't going to melt or anything, is it?
You'd have to work very hard to melt stainless steel, and it's certainly not going to happen if you've got liquid in the pot. The problem with thin stainless steel pots (or any pot) in direct contact with electric burners, or over really powerful flames, is that they can develop hot spots which can scorch your wort.
If you're an AG brewer with a relatively clear, clean wort, this isn't so much of a risk, but if you're an extract brewer or are adding adjuncts to your kettle, it's easier for the concentrated stuff on the bottom of the kettle to scorch. A lesser hassle is very strong direct heat can darken your wort, which is a potential problem if you're trying to brew a very light-colored beer. That's one of the reasons why homebrewed light lagers almost never come in as light as commercial versions.
If you have trouble with scorching, the quick solution is to put a "flame tamer" of some sort under your pot which acts as a heat diffuser.