I don't believe it is currently possible. The only app with access to the time/notification background api is the built-in calendar and the clock. Hopefully in future iOS versions, they will open this up.
In the mean time, you could try to figure out something with the built-in calendar and the rtm ical files. Good luck :-)

I don't think this is impossible. My previous task management program (Task PRO - which I abandoned due to poor support, gratuitous capitalisation and the fact they offered no synchronisation between iPhone and iPad) could do this fine.

It might have only worked if the application was running in the background, but certainly it could remind me at 2pm of a task being due without any internet involvement (as I said, they don't have a server to manage the tasks at all!)

Pretty sure HTML5 will save the day on this request. Depending on how the existing apps are built maybe they can implement HTML5 application cache as well. http://www.html5rocks.com/en/tutorials/appcache/beginner/