Right now, nothing from the App Store does true background notifications.
There are some options, but it depends to some extent on how fast you want the communication to be. You could use the Blackberry e-mail address for the BB (or any other e-mail address that pushes well) and a push e-mail account on the iPhone, like a Yahoo one. That way it should push on both sides, although it might not end up still being as quick as SMS.
Otherwise, you might explore the jailbreak type of options....