MMS and Restricting background data
Background data is anything that involves mobile data or Wi-Fi; it will use whichever is enabled (I think Wi-Fi has higher preference). This includes autosync, background polling, notifications, and other similar operations that will access mobile data.

Assuming you don't have any task / data killer that runs in the background when your phone sleeps, the MMS will be sent You do not need to have the Messaging app open. However, if the sending fails (after a certain number of retries), it will fail and you will have to resend it manually.

Update #1: Autosync depends on background data. If background data is disabled, autosync is also disabled. Autosync is used by accounts defined in settings (In S4, System Settings &gt; Accounts). 

BUT it's also possible to have background data enabled, and autosync disabled. In this case, apps could poll and show notifications (e.g. Twitter).