Two most well known area's in the vicinity are Chattanooga and the Red River Gorge. Big South Fork also has some climbing. Check the online guide here for routes. Check redriverclimbing.com for the Red.
As far as I know, there isn't anything within 45 minutes of Gatlinburg. Big south fork is about 2hrs away, as is the Obed. Devil's racetrack, near Caryville, is about 1.5hrs. Ijam's nature center, in South Knoxville, has a climbing area now - a handful of short but worthwhile sport routes. Ijam's crag would be about an hour from Gatlinburg, which may be the closest climbing to where you are.