I just got back from there last week, we spent a week in Gatlinburg. Took the bike, but one I got there I realized that I wanted to live so it never got assembled.
There is no, repeat no, off road riding allowed anywhere in the Great Smokie Mountain National Park, but I knew that going in, I figured I would ride the roads on my MTB. Again I wanted to live so no dice, there is no room for both bikes and cars on those roads, and as I drove the roads it became very clear that I had made the right decision.
Of note: Each Saturday and Wednesday they close The Cades Cove loop to cars from sunrise to 10:00. I'ts paved, very flat, and far, about an hour a half from Pigeon Forge, but supposed to be scenic.
Best bet, make the trek to NC or GA.
I hope this helps.