Initially I have two thoughts.
First of all, if you rent through Airbnb then obviously you pay with a credit card. This isn't so applicable to a current living situation and is somewhat less flexible, but it does work. I rented a place on Airbnb for 3 months this summer up in Boston.
My second thought is that I doubt there would be any way to do this in an economically efficient way because the credit card points, at least to some degree, come out of the fees that credit card companies charge for businesses (or in this case landlords) to accept credit cards. The point being that to use your credit card to pay rent, your landlord would have to pay a 2-3% fee to the credit card company, which they would most likely roll into your monthly rent. So in the end you're paying 2-3% more in rent to get 1-2% back in points on your card. Its possible you could find a way to do this temporarily, but economically speaking, it would be inefficient and the gains wouldn't last long because someone would figure out they were losing money.
I'd definitely be interested to hear other peoples' opinions.