The best thing to do for that would be to have a Playlist of the videos you wanted to feature.  You can have a playlist of any videos as long as they allow it in their settings.  Unfortunately, that’s a paid option in Involver.  
This one also looks like a paid app:  http://apps.facebook.com/tabappforyoutube/This list is hard to tell which apps might be paid:  http://www.reelseo.com/facebook-video-apps/#youtube

This app looks like it might work for you – I think it selects favorited videos but check it out:  http://apps.facebook.com/videobox/