Does your phone have to be paid off to trade-in for a newer iPhone?

Depends - is this iPhone enrolled in the Apple iPhone upgrade program? If so, do you meet the terms for upgrading? Have you made 12 payments? - iPhone Upgrade Program - Apple

yes or it gets added to the new phone.

Then you will have to pay it off before you can trade it in with Apple.

You may be eligible to trade up on your carrier's upgrade program earlier without paying it off. Check with them.