I don't believe allergies in amphibians have been studied. I've never encountered anything I could say was such, and I've never heard hobbyists report it.

They can get sick, and that's a whole encyclopaedia right there. The most common disease is bacterial septicemia brought on by stressful conditions. It can be caused by a number of different pathogens. Treating it is difficult but usually involves administration of antibiotics by a skilled professional (a veterinarian).