All instructions I have read say to rinse. I have brined 4 or 5 turkeys and never rinsed them. I kind of held them up and let as much as I could drip off and then pat dry with paper towels. Never had an issue with too much salt or anything.
We brine then smoke our turkeys. We never rinse them off. The brine is mostly water anyway. Rinsing is kind of unnecessary in my opinion. We dry our birds off well and put them in the fridge uncovered to dry out the skin (for smoking). Always good!