Fungal infections such as ringworm can transmit across species. Ringworm is, I believe, rare in dogs and cats, but horses are at fairly high risk and it's not uncommon for somebody treating ringworm in a horse to catch it themselves if they aren't careful. (Ringworm is a skin infection, it's not serious but is quite annoying).

Another disease that does transmit that really should be mentioned is bacterial conjunctivitis. Bacterial conjunctivitis can be caught by and from humans, dogs, and horses. However, the same is not true for cats. So if your dog or horse has pink eye be careful about touching their eyes and wear gloves while treating them. A few years ago a horse at the barn I ride at caught pink eye from a child. Viral conjunctivitis is another matter, but if you have pink eye and don't know which kind it is yet, try not to pet your dog until you're sure...and have been on antibiotics for a day or so if it IS bacterial.