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Good question! None of us knew the answer (because it's up to the iPhone whether or not to keep running OmniFocus in the background), so I just experimented with this while watching my sync server's web logs.

At the moment, it looks like OmniFocus won't start any new syncs after the screen is turned off, but if you already have a sync in progress it will continue to run in the background until it finishes. Note that this could change in an iPhone update if Apple changes its policy about how apps run while the screen is off.

Also, I should note that that only applies to the top button; if you hit the Home button in the front, OmniFocus will immediately terminate and abandon any sync in progress. (It will pick up where it left off the next time you start it.)