LogMeIn Ignition has just been updated for iPhone and iPad specifically with iOS 4 support.
Normally, this would not necessarily be too exciting, except I just got done testing this release. I am very excited to report that I was able to log in to my remote desktop, then check mail and other apps. Put the iPhone to sleep, come back minutes later and open LogMeIn Ignition and... my desktop was still there!
They're no doubt using iOS 4 multitasking's task completion feature (the same one that lets uploads finish in the background). It won't let you keep the connection open forever -- more like 10 minutes -- but it's still fantastic and I can't recommend this app enough. This is very exciting to see developers adopting iOS 4 so quickly and in a useful manor.
Oh, and here is a tip. If you keep LogMeIn "running" in the background for a while, and want to make sure you don't get a timeout error when you return to the app, check the setting on the client machine. It needs to be more than iOS 4's 10 minute task completion limit (the default on mine was 20 min.)
[$29.99- iTunes Link]