The mb.YTPlayer doesn’t work on any mobile devices (iOs, Android, Windows, etc.) due to restrictions applied by the vendors on media controls via javascript. Adding a background image to the body as mobile devices fallback is a good practice and it will also prevent unwanted white flickering on desktop browsers when the video is buffering. The theme does display a placeholder image in mobile devices.
There are some alternate solutions provided in these threads –
http://support.livemeshthemes.com/forums/topic/video-issue-on-mobile-device-and-ie-8-9-10/#post-8518
http://support.livemeshthemes.com/forums/topic/video-when-i-visit-the-page-from-iphone/#post-6083
Hope this helps.