Apps don't run in the background. When you do move away from them they tombstone, or go into non-active state, using virtually 0 power. So only the current active app, behind the lock screen, can consume power when active, like a radio app for example.
FYI, live tile apps don't check for updates. They can't because they don't run in the background. But your phone always tries to maintain a connection to the data network in some way. When new info is available it gets pushed to your phone. So push email is actually more power efficient than checking ever 15/30 minutes for example as the low energy connection is always there anyway. If you have access, you can turn off 3G (but leave data on) when you don't need a high speed connection for streaming or downloading large chunks of data. That will save significant battery power.
I really want to look into that more. Because that is the case with push on BlackBerry devices as it's being pushed from RIM's NOC when data is available.
On the iPhone for example, push uses more battery. This is something even Apple has stated. I believe that is the case for Android and possibly Windows Phone.
I believe it is because the devices has to keep a connection (secure) to the server.
Found this on battery saving tips on apple's site:
Turn off push notifications: Some applications from the App Store use the Apple Push Notification service to alert you of new data. Applications that extensively rely on push notifications (such as instant messaging applications) may impact battery life. To disable push notifications, go to Settings > Notifications and set Notifications to Off. Note that this does not prevent new data from being received when the application is opened. Also, the Notifications setting will not be visible if you do not have any applications installed that support push notifications.
Fetch new data less frequently: Applications such as Mail can be set to fetch data wirelessly at specific intervals. The more frequently email or other data is fetched, the quicker your battery may drain. To fetch new data manually, from the Home screen choose Settings > Mail, Contacts, Calendars > Fetch New Data and tap Manually. To increase the fetch interval, go to Settings > Mail, Contacts, Calendars > Fetch New Data and tap Hourly. Note that this is a global setting and applies to all applications that do not support push services.
Turn off push mail: If you have a push mail account such as Yahoo!, MobileMe, or Microsoft Exchange, turn off push when you don?t need it. Go to Settings > Mail, Contacts, Calendars > Fetch New Data and set Push to Off. Messages sent to your push email accounts will now be received on your phone based on the global Fetch setting rather than as they arrive.
Auto-check fewer email accounts: You can save power by checking fewer email accounts. This can be accomplished by turning off an email account or by deleting it. To turn off an account, go to Settings > Mail, Contacts, Calendars, choose an email account, and set Account to Off. To remove an account, go to Settings > Mail, Contacts, Calendars, choose an email account, and tap Delete Account.
Another thing to keep in mind is the intervals it's fetching data in... so if it's checking every 10 minutes, but you receive on average 1 email every 2 hours, you're using more battery life that isn't needed.
Exactly. If you typically get emails every few minutes, push usage might be extensive, but if you only get a dozen push messages throughout the day then checking every hour is probably less efficient. WP7 maintains a constant connection to your carriers data channel to receive push and send push requests. They talked about this in the MIX11 Live Tile update. The connection uses minimal energy when idle, but less than having to fire things up and search for a connection every time. Also, it's sort of necessary for the push/live tile system to work.