iPhone upgrade program

You can trade in the phone for a price reduction on a new one, but that's not going to do anything to whatever you owe to your carrier. You still have to pay that off.

Apple's iPhone Upgrade program, and the Apple Trade In, are completely independent of your carrier and whatever you owe to them.

Nothing. If you still owe money to your carrier, you need to pay that off. Apple's iPhone upgrade Program has no connection to your carriers plan so will not affect it at all in any way.