We almost certainly can’t catch a disease from them in the same way we can’t catch most animal and plant diseases. Each virus and bacterium relies on certain aspects of the host to survive and reproduce.

Fungal infections are more possible, because fungi just feed on decomposing organic matter. As long as they’re carbon-based, they’re probably edible. Same for parasites: as long as they’re made of meat, they’re edible. (Barring any strange chemical composition, like if for some reason they carry large amounts of what we would consider toxic metals.)