I actually found an old post that solved this issue. In addition to the instructions, you also have to click the 'include music video' box in the sync music section.
IOS 6 - Listening to Videos with the screen off or in the background??
JorrymeOct 13, 2013 11:41 AMin response to greencore
Looks like this is not yet solved in ios7.
Another workaround - tested on iphone 3gs, iphone 5 and ipod 4 on ios6; ipod 5 on ios7
(requires itunes)
1. Choose video(s) you want to background play - or any you would like to have in a separate playlist for that matter. **I tried this with an itunes purchased tv show as well as dvds I've ripped that have no drm and downloaded and converted youtube and other sources. All worked for me on current (11.1.0.126) and past itunes editions.
2. Right click and choose "get info" (yes, you want to edit multiples).
3. Click on the tab "options"
4. Change media kind to "music video"
5. All videos you chose before clicking "get info" now disappear and reappear in "Music"
6. Go to "Music." Make a playlist and sync to your device or copy these music videos however you usually copy music to your device.
7. On your device, look for the videos in your music app. If you didn't already use a playlist, you can now put them into playlists if you want to.
8. To background sound play, start playing the video of your choice (within the music app), lock your device (it will stop), double tap home button, press play ("music" will start), and lock phone again. Will continue to play. Alternatively, after starting the video, you can press the home button, then double tap the home button, scroll left to right, press play, and you can use other apps while the sound of the video plays.
