Kimpton Hotels have the best pet policy anywhere -- no size limit and no dog fee. Kimpton says, "pets are welcome regardless of size, weight, kind, or breed". They can be pricey, but they run frequent specials.
Other hotels where we have stayed that allow dogs (and that my parents and I recommend) are:
Homewood Suites by Hilton Stratford, Stratford, Connecticut (website)
Aloft Baltimore Washington International, Linthicum, Maryland (aloftbwi.com)
Stonewall Jackson Hotel & Conference Center, Staunton, Virginia
Westin Washington Dulles Airport, Herndon, Virginia (they have a 50 pound weight limit and a 3 page waiver to sign, but there's good walking nearby)
Sheraton Premier Tyson's Corner (no dog fee, no weight limit, welcoming and friendly staff)
1900 Inn at Montford, Asheville, North Carolina
Holiday Inn Express, Dillsboro, North Carolina
Holiday Inn Express, Morehead City, North Carolina (dog-friendly and a short walk from a very nice neighborhood for walking, which to us is a real plus)
Beaufort, South Carolina is a wonderful place for a pet-friendly vacation. Check out Pet Friendly Beaufort for more info.
City Loft Hotel, Beaufort, South Carolina
Tides Folly Beach, Folly Beach, South Carolina
Hampton Inn Savannah I-95 North, Port Wentworth, Georgia IS NO LONGER DOG-FRIENDLY :(
Aloft Jacksonville Airport
Hilton Melbourne Beach Oceanfront, Melbourne Beach, Florida
Vero Beach Hotel and Spa (Kimpton), Vero Beach, Florida (Dogs aren't allowed on the beach in Vero Beach, by county ordinance, but there are plenty of sidewalks and a dog park nearby.)
Hilton Key Largo Resort, Key Largo, Florida (Dogs aren't allowed on the hotel's private beach.)
Casa Morada, Islamorada (Dogs are not allowed on island with pool, but are allowed everywhere else on the property.)
The Westin Key West Resort and Marina, Key West, Florida (Very nice and super convenient!)
My mom's reviews of these places can be found on Trip Advisor under Labnut or Garth Riley (Labnut - Member Reviews).