Technically, you could have an app that plays silent audio in the background to keep from getting killed and then do something else, but that would get flagged by the review process.
But, if I recall correctly, VLC will continue a background file transfer session from its built in web server on iOS if you play a movie in the background. Otherwise it will be killed after 10 minutes.
I think they've wisened up to that trick. My girlfriend and I use a guided meditation app that has long silences in the audio. If the screen is off, her phone kills the app after about thirty seconds of silence, making the app practically useless.
I was going to report a bug to the app suggesting they include some ultrasonic or subsonic audio to defeat this mechanism, but it's just an arms race at that point. The phone will start looking at the total power in the audible spectrum to decide whether to kill the app or something.
This kind of thing is infuriating and prevents apps from doing anything outside of the box, stifling innovation and making users frustrated.
But, if I recall correctly, VLC will continue a background file transfer session from its built in web server on iOS if you play a movie in the background. Otherwise it will be killed after 10 minutes.