Can you upgrade your tires from 31x10.50R15LT to 31x12.50R15LT without anything happening?

Depending on the type of truck and lift, the 12.5 can be "too" fat in appearance and might rub on the interior wall of the fendor. You might try or look at the 11.5. The 12.5 is very good in loose dirt and sand, but will not do as well on harder terrain.