Some are run by airport volunteers and are marked that they are not official post office boxes. It's possible that the JFK one is no longer collected. They should weld the slot shut if that's true. I wouldn't chance it.

If you want to use that mail slot a lot, you could mail yourself a post card or letter and see. It would only be worthwhile if you travel to JFK often. An alternative is for you to mail something that you anticipate getting a response and can resend it.