Depends a little bit on the tire. If you are at stock ride height, you might get a little rub on the front mud flaps at full-lock turn when the front suspension is compressed.
I had a Toyo AT tire 265/70/17 when I was at stock height, and that is the first place they will rub. These tires have a rather square outer edge on them, so if your potential new tire has a rounder edge, then you might not see any rub at all.
I had a set of Michelin LTX M/S2 265/70/r17 on my 2003 Sequoia Limited. At full lock, they would rub on the frame. Not a big deal, though; wore off maybe a couple square inches of paint on the frame, but no damage.
These are Michelin Defender 245/65/17 tires on Toyota split spoke wheels. These are a bit smaller than what you're asking about but work well on my rig.
No rubbing issues to date.
I am running 285/75R16 on my 2004 SR5 Double Cab. So my tire size converts to 33" x 12.50" vs your size converts to 31.6" x 10.4". I do have a leveling kit on the front though. These tires rub on the frame at full lock, but no big deal, either don't turn at full lock or just let it rub.
I am running no name 265/70/17 highway tires and no rub whatsoever on my '06 DC. Just lifted the front 1" with some Billy 6112s in preparation for some taller tires though.
Are you running stock rims? Mine are stock if that helps.