Most of the major issues have been brought up here. Emma makes the climb with us every year for STP, but she also climbs just about every weekend all year long and usually makes several trips above treeline each year.
Things to consider are:
Recently a friend of ours took her dog above treeline on Mounts Madison and Adams. The dog tore her paw-pads significantly. The rocks above treeline in the Northern Presidentials are made up of quartzite and mica-schist. This is extremely rough as anyone who has taken a tumble on it can tell you. Imagine hiking barefoot across this. That is what a dog does.
On a trip 2 Saturdays ago above treeline Emma overheated and we had to cut the trip short. Above treeline there is no place to get out of the sun. It can be like crossing a desert.
Old dogs may find the trip too much. Young dogs who are new to this type of hiking cannot pace themselves and run mad until they bonk. Also, as Jim points out, dogs who run off trail not only tear up the alpine flora, they put themselves at risk of running off a cliff. Many dogs have gotten themselves hurt or killed here, so have people, the difference is the people had a choice, the dog just wants to please its master.
Dogs pull muscles and get sprains just like we do. Tying up a rescue crew for a dog when there are people who may need them is not considered good form.
Not everyone likes dogs. Even a dog as innocuous as Emma can be a concern to some people.
If the dog bonks at the summit and you decide to take the Cog, they won't let the dog on. Not sure about the shuttles. If you want to tour the Obs while you're there I'm not sure they'll let you in with your dog. Emma has been in, but she has friends there. At least she did, most have moved on including Nin.
I guess this all sounds as though I'm trying to dissuade anyone from bringing their dog up MW. I'm really not, but I want people to consider what the dog will experience. Someone who has never climbed MW before should go first and see what they think before bringing a dog. Maybe they will think "This is nothing, I could have brought my dog", but it's also possible they will think, "Thank God I didn't put Rover through this!".