Generally speaking the dog friendly sections as I know them are:
1) Green River between Green River and Mineral Bottom;
2) Colorado River between Loma and Westwater, limit 2 and they count toward your total trip number (i.e. 23 peeps and 2 dogs is the max);
3) Dolores, all sections below the Bradfield launch;
4) Colorado River between Cisco Landing (Westwater takeout) and Moab;
5) Colorado River Pumphouse run
7) Salmon, certain sections;
Escalante River (Utah, not the creek on the Uncompagre Plateau).
I know that you can't have dogs on Westwater, Desolation and Gray's, the San Juan below Sand Wash, Gates of Lodore, Stillwater Canyon, Cataract or the Colorado below Potash. My information on the Salt and the Salmon is 2d hand. I don't know about the Rio Chama, the Rio Grande (any section), the Smith in Montana, the Gila in NM, or any of the rivers in the Northwest. I have assumed the Escalante is dog friendly because a friend took their dog, but I could be very wrong and they simply didn't get caught.
I'd like to know if anyone has any more specific info about Dog Friendly rivers. The best trip I ever took with my dog was with two buddies and my lab in open boats on the Dolores between Slickrock and Bedrock.
Be smart, harnesses with a backstrap make it easier to pick your BooBoo up like a wet sixpack and haul her/him back into the boat, and life jackets are a really good idea in whitewater. Swims can be really long and don't plan on getting your dog into the boat quickly.