I had never come across this before or even considered it however, after some research I came across a paper that describes human to animal transmission of Staphylococcus aureus, influenza A virus, Cryptosporidium parvum (protozoa) , and Ascaris lumbricoides (round worm) into various animals.
While the researchers say they have found these human diseases in animals they are less sure about how the process actually occurs.
But these are organisms that do infect humans so my conclusion is that maybe there are no diseases we are immune to that infect animals. In fact this would make sense as if we were immune to a particular pathogen then we would not detect it. Whereas other animals who are susceptible would show signs of the infection.