DVDs are read by a laser, so they never wear out from being played since nothing touches the disc. Pressed discs (the kind that movies come on) will probably last longer than you will, anywhere from 50 to 300 years.
Expected longevity of dye-based DVD-R and DVD+R discs is anywhere from 20 to 250 years, about as long as CD-R discs. Some dye formulations (such as phthalocyanine and azo) are more stable and are expected to last longer, 100 years or more, compared to 20 or 30 years for less stable dyes. The phase-change erasable formats (DVD-RAM, DVD-RW, and DVD+RW) have an expected lifetime of 25 to 100 years.
In actuality DVDs often don’t last as long as the above statistically-based longevity figures would lead you to expect. Longevity can be reduced by problems in manufacturing or recording and by poor quality material. Shoddy pressed DVDs may deteriorate within a few years and cheap recordable DVDs may produce errors when recording or may become unreadable after a while. In other words, you get what you pay for. If longevity is important, invest in high-quality media.
In 2009, Millenniata introduced M-ARC technology (also marketed by Cranberry as DiamonDisc), a DVD-compatible recordable format using an obsidian-like synthetic recording layer etched by a high-powered laser. The discs are expected to last hundreds of years. They require special, very expensive recorders, but the discs can be read by standard DVD readers. Because of the high recording costs, Cranberry offers an online service for uploading data to be burned.
For more info on disc longevity see The Relative Stabilities of Optical Disc Formats, Lifetime of KODAK CD-R Ultima Media, and Professor Kelin J. Kuhn lecture notes.
For comparison, magnetic media (tapes and disks) last 10 to 30 years; high-quality, acid-neutral paper can last 100 years or longer; and archival-quality microfilm is projected to last 300 years or more. Note that computer storage media often becomes technically obsolete within 20 to 30 years, long before it physically deteriorates. In other words, before the media becomes unviable it may become difficult or impossible to find equipment that can read it. Optical media is proving to be one of the exceptions to this rule, since DVD and Blu-ray readers can read CDs from 1983, although the CD/DVD/BD format will begin to fall out of use around 2020.