Not sure the screen has to be on exactly, but definitely when I had an iPhone 5 (pre iOS7 7) it would interrupt the uploads at some stage during the process of turning the screen off, locking etc. You'd get a notification about uploading being paused. Android does similar things but mainly (only?) when battery is low.

I don't have the iPhone any more so don't remember the exact mechanics, but it does happen. My wife's iPhone 5 does it with iOS7. I know she puts our kids photos in a special Dropbox folder for relatives to see and sometimes they don't upload until she has enough screen on time it seems.