I was wondering if I could use a 31x10.5 tire on a 15 x 6.5" rim, considering that I'd be airing down frequently. The tires I'd be using are BFG AT/KOs, which have a pretty stiff sidewall, and at that size they recommend 7" rims. These would be used offroad and not much on road except for towing.

dont know about airing down but i run 31x10.5's on stock 5.5 rims no problem. i however dont really have a need to air down when im out wheeling. but i have towed with them and have no problem other then im lacking a bit of gearing on the hills.

Actualy running a 10.50 tire on a 6.50 rim will work well when aired down. You will be less likely to throw a bead . I wouldn't recomend running them over 20 lbs when aired up though. It could cause excesive center wear......Boxcar...

I run 11.5 X 32 on 7" rims and never had an issue, going down as low as 15 lbs and as high as 32. At first it was hard finding the right tire pressure to get the correct footprint and ride quality, but eventually I found it. I also like running the wider tire on a narrower rim because the sidewalls buldge out more protecting the rim a bit. This is kinda important to me because I run aluminum rims, with steel it's not so much of an issue.