The master Genius playlist does not automatically sync. There is a workaround.
If you want to sync the variable Genius playlist to the phone create a new Smart Playlist with the rule: <Playlist> <is> <Genius>
Name it whatever, make sure you checkmark it on the iphone's music tab if you only sync certain playlists, and you're ready to go.
It is safe to leave Live Updating on. The base song that the playlist is built on is not cleared when you exit iTunes, so the playlist stays the same until you choose a different song... Or until iTunes does an auto-reprocessing of your library I suppose.