I believe the problem is that somehow the iOS device is losing connection with iCloud and for whatever reason when this happens it defaults to turning off all iCloud services. This is very easy to see on my iPad 2 when I go into Settings after I haven't been in there awhile and going to the iCloud settings. The Find My iPhone setting will be off, but will change to on in a few seconds. It's off because the device doesn't seem to remember the state of Find My iPhone and needs to ask the server.
At the same time my iPad makes a ton of connections to Apple's servers as if iCloud was just enabled or airplane mode was turned off. My iPad 2 has 3G so it's always online, meaning something is screwed up with the device losing connection to iCloud and having to re-establish. Speaking of airplane mode, turning that on and off causes my iPad to simply drop connection to iCloud such that it prompts for the iCloud password when I disable airplane mode.
What's odd though is that if I use Find My IPhone to find my iPad, it works, except when Find My iPhone gets permanently turned off. In that case it says my iPad is offline until I re-enable it on my iPad. I'm convinced this is an iOS 7 bug, despite only seeing it on my iPad and not my iPhone. On my iPad it happens about once a week requiring a restart to get iCloud working correctly and then requiring me to re-enable Find My IPhone.
Unfortunately my iPad is out of warranty so I can't report the problem to Apple.