Haligator, the place a bike is most worthwhile is Penns at Poe Paddy. And for a while it was real nice to have a bike there. A hiking/biking trail goes through the tunnel at Poe Paddy and parallels the stream all the way down to Cherry Run. A bike made it easy in/easy out to the middle of that area.
But the tunnel is currently closed. So to access it from Poe Paddy, you have to go over a fairly steep mountain spur to get to the trail. I'd think the hassle of lugging a bike over would be more than the hassle you're saving once on the other side. And keep in mind you'd most likely have to lug it back over to the car side, at night.
A bike would still be nice if parking at the Cherry Run end, but that's a 1+ hour drive, vs. 25 minutes for the Poe Paddy end. Again, I'd question the usefulness of spending an extra 30+ minutes on the road to save a little walking.
Another spot is Spring Creek, in the canyon. A bike would be nice in there.
Other than that, I think it's probably more hassle than it's worth. Anywhere you could use the bike, you could also use a car. For instance, Spring Creek below Paradise. Nice road for biking. But you can also drive it.