I think your main concern is how to receive new messages while your application is in the background. Because it's not like I'm going to have a messenger app all day open when I can get notifications with apps like WhatsApp.
Better than TCPIP sockets you could use websockets. Since it is HTTP there are no firewall problems, BUT that requires a permanent connection with application on the foreground which drains the battery.
And because only music, location, or voice ip, is allowed to run on the background, you can't poll unless the application is open. Note that if you register for those background tasks and do something else Apple will notice and reject your app.
Therefore, use APNS. That's what WhatsApp does.
