My dog(s) have always been welcome at tent platforms in the Whites, and in shelters and tentsites on the Long Trail in Vermont and Maine/Mass/NY/CT AT, in the Provincial Parks of Nova Scotia, the Adirondacks and three of Colorado's Wilderness Areas. I carry a rubber coated ten foot cable tie out with snap closures on both ends for tethering to guard against my line getting chewed or destroyed by campground mice, etc.
We have found the sites in the WMNF to be dog friendly, note that in the more popular sites where a caretaker is present, some tent platforms are not private. One example is Garfield Tentsite, where several campers from separate groups may be fit together on one platform by the caretaker as the tentsite fills up. During the busy season along the AT corridor, we try to defer to thru hikers and for this reason carry a tent and use backcountry/stealth sites.
As always, checking ahead with the agency managing the land is the best way to assure dog friendliness. I also find its helpful to include contacts for local veterinarians when planning extended backpacks with canines.