Nope. Animals are less susceptible to the diseases in their prey, IF the prey they eat are known to have certain diseases.
A better way to explain this would be in certain regions humans are practically immune to diseases, that outsiders are not. Its mainly because those humans are less likely to catch the disease that their body is native to. But lets say an American goes into that area. The American would be more susceptible into getting that disease, because the disease isn't always affecting us.
hope that makes sense.
So your animal question would be more about if the species they are eating is known for a certain pathogen. In coevolution, the predator would most likely develop an immunity to that pathogen and is less susceptible into catching it