I'd normally say yes, but if your surface prep is flawless (no rust, no pits, and smooth) you don't *have* to prime it.
Besides with a coin door the thinner the application on the hinges and edges the better. That is why I usually use satin black (at least on those portions).
Oh and don't forget to bend/rubber mallet the door a bit if needed to make sure it opens/closes smoothly before you paint. Always sucks to get it painted and reassembled and then remember that it rubs somewhere when you close it. Makes the whole process a waste of time.