[Andrew]

Only certain kinds of apps / tasks can run in the background. (Music apps and some location-based tasks, for examples.) OmniFocus can (generally) sync up any changes it has made in a few situations: periodically, while running in the foreground; in the background right after switching to another app; and in the background right after the iPhone has been put to sleep. But it has no way of periodically downloading updates from the sync server in other situations, sorry.