IOS – Push notifications work everywhere but the home network

Have a look at the ports commonly used by Apple (on all devices, not only iOS) here (Though these are only the ones used by Apple and other Developers might use other ports) and look for specific Services (FaceTime uses many ports, so does Game Center and so on…)

For what it's worth, my iPod Touches' (2g and 3g) WiFi remains on when the screen is off, and I do get push notifications from various things (in fact, as I was typing this I got one from Ambiance). If you go out of range of known networks and "Ask to join networks" is turned on in the WiFi settings, you of course won't get push notifications unless you turn the screen on and accept an open network. The Apple battery life optimization page has a little more information, including how to control on a per-application basis what push notifications will work.