Dogs are allowed to hike with you on parts of the Appalachian Trail so you can take your dog with you when hiking the Appalachian Trail.
However there are a few places along the Appalachian Trail where dogs are not allowed to hike through or stay at with you such as the Smoky Mountains and Baxter State Park.
Those are the only two places that dogs are not allowed when hiking the Appalachian Trail so if you want to visit those places along the Appalachian Trail it would be best to not take your dog with you when hiking the Appalachian Trail.
But as long as you avoid the Smoky Mountains and Baxter State Park you can take your dog along with when hiking on the Appalachian Trail.