...or debit card to pay rent? Is there a limit on convenience fees? Does it vary from state to state?

I am pretty sure your landlord is violating thier visa/mc/amex/disc contracts by charging a fee. I think merchants are not allowed to penalize credit card customers for using a credit card vs. cash or check.

Your landlord can charge any fee he likes as long as you signed it as part of your lease. Check your paperwork to find out if he states that he will do this. ~*~