With the app closed? None exists.
Some apps autosync on opening the app. Pocket Informant does that.
I just checked, the lite version doesn't have all the autosync options that the full version does. You'll find an AutoSync 'At Startup Only' option in the full version.
The only other apps I know do that are Todo and 2do, but you've already said Todo isn't what you're looking for, and 2Do is a similar set up to Todo.
Have you checked out Week Calendar? It allows you to sync all of your calendars together. If you use Toodledo (like I do) then it's possible to add Toodledo as a subscribed calendar. Once you add tasks through Toodledo, your native calendar will pick it up and sync with Toodledo. This way you'll have your tasks implemented into your calendar.
This should work with Week Calendar as well. Week Calendar should sync up with all calendars that you have on your iPhone.
However, I'm having trouble with exactly that. Whenever I add or edit a task through Toodledo, Week Calendar does not pick it up at all. The only way for me to see my Toodledo tasks in Week Calendar, is by starting up the native iPhone calendar first, allowing it to sync, and once i can see the tasks in the native calendar I'll be able to see them in Week Calendar as well. There is no other way for me and it's quite frustrating. Does anyone have any idea how to fix this? I e-mailed Wasabi-Apps, but no solution came from that.
It seems to sync every time you open up the Calendar app, the little spinner indicating an internet connection in the top bar (the one with the time) always appears whenever you open up the calendar app, it seems to take some time for the calendar app to process the info and for it to appear in your calendar though.
Hmmm... not my experience. PI always takes a minute to sync and all new events show immediately after that, but it at least takes the calendar app a a couple hours for new events to appear from Gcal to the calendar app. Perhaps that's just cause I've got the Calendar app set up to sync with Gcal via CalDav.ya, but it takes about just as much time to sync as PI.
Hmmm... not my experience. PI always takes a minute to sync and all new events show immediately after that, but it at least takes the calendar app a a couple hours for new events to appear from Gcal to the calendar app. Perhaps that's just cause I've got the Calendar app set up to sync with Gcal via CalDav.
How's Week Calendar if you bought it?