Only a couple of weeks ago I was asked what the rules were for dogs on the Tube. As far as I know "carrying dogs on escalators" is the only rule.
Oh, and I don't think that dogs are allowed to sit on the seats, although to be honest I've never seen any signs to that effect. Otherwise you don't have to pay for dogs to travel on the Tube. However, if they are travelling by themselves they may be approached for Oystercards or a fare! Feel free to jump in if I've left out any other Tube rules for canines.
Finally, here's a picture from my archives, sent in by dog fan, Frank Timmer, of a man failing miserably with the "Dogs must be carried" rule.