Other humans followed by non-human primates and dogs and cats iirc. Basically the more closely related it is to humans, the more diseases it can give them (and vice versa since humans can give many diseases to non-human primates, for example). Anything that's not a mammal isn't going to be able to give as many diseases as mammals.
I know most cases of rabies in humans are actually caused by feral cats and dogs, particularly in developing countries (India has a bunch, which is related to the Indian vulture crisis causing an explosion in the feral dog population).
Here's a
paper on diseases dogs can transmit and some
from cats.
There's a reason why bites from cats and dogs are considered the worst non-venomous animal bites you can receive after bites from humans.