If you happen to find a dead bird in particular, please don't pick it up. They have mites on their feathers (skin?) that will absolutely jump on you and absolutely fuck up your skin. You can literally jump in the shower once the itching starts, and you will be in for something like localized poison ivy where those little microscopic (probably not, but they were so tiny I couldn't see them) assholes were, or at least wherever they bit you.

I would generally caution against actually touching anything that is dead. Too many pathogens, nasty bacteria, and potential touch contracted illnesses.