One of the ones that always struck me was Gatlinburg, Tenn., and Cherokee, NC. It's a 33-mile drive across Newfound Gap Parkway in GSMNP, but that's closed to commercial vehicles. I never bothered to figure out how long the detour would be, but it would involve US 321, I-40, US 276 and US 19.
In Kentucky, water has created a few issues where portions of counties are cut off from the rest of the county. There are a couple of instances along Lake Cumberland, plus the infamous Kentucky Bend in the far western part of the state.