If you touch a dead bird with your bare hands then you might contract some diseases from the dead bird which could make you very sick or even cause fatal diseases.
If you need to touch a dead bird you should wear rubber gloves or other gloves to prevent getting infected with any diseases that the dead bird might have.
Or just use a shovel to remove the dead bird and either throw the dead bird in the trash or dig a hole and bury the dead bird.
Wash your hands with antibacterial soap and hot water to kill off any germs that may have gotten on your hands.