I found a way to fix this problem on mine (iPhone on iOS 7.1.2)
Keep your music videos as 'Music Videos' (meaning, don't change them to TV Show as suggested earlier by some in this thread), then do the ff.:
1. On iTunes, go to Music > Playlists > Music Videos
2. Select all files > Get Info > Click Options > Change Media Kind from Music to Music Videos*
3. In Settings > Video, make sure to set to 'Start from the beginning'
4. Sync your iPhone. iTunes should update all the music video files on your iPhone and VOILA! They should now play continuously.
*If you check the files individually, it will say Music Video under Media Kind. But for some reason when you select/highlight all, it says Music.
Hope this helps!