Physical Danger
Um...yeah, there's danger. You're working with wild animals, after all.
No one's going to send you to save a wounded bobcat without proper training, but even if you're just volunteering at your local shelter, you will be feeding and treating animals in cages—and these animals do not want to be caged.
A caged hawk is going to try to bite you. No matter how many times you call him Mr. Freckles, he is not going to be your friend, and he will make that abundantly clear.
Out in the wild, the safety risks increase considerably. You're going into dangerous areas, at all times of day, in all kinds of weather. You'll have gloves and equipment, yes, but the animal is still going to resist when you get close.
There is also the very real chance that you will be exposed to animal-borne diseases. You will need to get vaccinated, and you will need to be careful to ensure you are working in a sterile environment and performing proper clean up.
All that said, wildlife rescuing is not quite deep sea welding—you're probably not risking your life every day. But you will be risking your nice, scratch-free arms and face...and don't get too attached to any of your clothing.