You can have your app continue to run for up to 10 minutes after it's been closed, so it should be possible to use that time to keep your connection open. If no one has contacted you in 10 minutes then chances are they wont in the next 10 either so it would be better to switch to push notification and let the phone manage notifications.

Keeping SSH or IM connections open is an action of a well known use for servers; listening for updates. And that's covered in the API.