Most of Pigeon Forge is easy to cover with a leashed dog at your side. Public outdoor areas are open to pets, and a leash is expected.
City parks are the obvious starting point. Patriot Park has a half-mile loop along the river that suits a short morning walk. Open-air shopping and entertainment areas welcome leashed dogs outdoors, but individual restaurants and shops set their own rules about pets coming inside. Ask at the door rather than assuming.
Bring waste bags and a collapsible water bowl. Both are easy to forget and hard to replace once you are out for the day.
Wherever you start, it helps to sort lodging first. Everything else is easier once your dog has somewhere to sleep.