You'll have to check with the bylaws of the town. For my town we are allowed but you must follow bike rules and clean up after the horse. It differs place to place, but if you find out that you indeed can, I'd recommend bringing along a letter from your officials or a copy of the written law stating that it is legal because some people surprisingly get defensive when they see horses on the road.
Its possible to hire horse drawn gypsy caravans for self drive holidays in Ireland - Horses and carts are a common sight in most parts of that country so I doubt you'll have any problems - always check first though as its the polite thing to do and remember you will be driving on the opposite side of the road to the US!!!