Let's see how long it takes to delete a video on Vine. This video was posted to Vine and then immediately deleted. Within the Vine app the post disappeared immediately, but the Twitter link took another hour to become inoperable: https://vine.co/v/bX1eOBrOHM5. Worst yet, the embedded video file is still available as is the thumbnail image. For the moment it seems that video posts on Vine are permanent, even if deleted.
Update 29 April 2013: 60 days later the thumbnail image is now deleted, the video remains.
Update 28 February 2014: Happy 1st birthday, deleted Vine.
Update 28 February 2015: Happy 2nd birthday, deleted Vine.
Update 28 February 2016: Happy 3rd birthday, deleted Vine.