I would not bother archiving the mainstream releases that can already be found on many torrent sites (like, you don't need to archive Star Wars or Lord of the Rings) and focus on the bootleg disks first. You just need any standard DVD drive, then use Handbreak to rip the disks to a video file. For official releases, many of them have forms of copy protection, but 15 minutes on Google should tell you how to get around any you come across.
Also, for reference, "burning" a DVD is writing data to a disk, so the opposite of what you're trying to do.