A mod company known as Team Symbiote has released an app for current Wii mod chip users to play DVD's on their modded Wii consoles.
While this isn't the first DVD playback solution for Wii owners, it does highlight a major issue with the game console and one that is really only a software limitation, so it is strange that Nintendo has not made available an official release that would allow owners play DVDs.
Instructions and further discussion are available on nintendo-scene.com forums. Simply burn with clone CD, Boot in modded wii with homebrew based mod programmed with YOASM 1.8. When menu comes up, hit 'z' to unmount, change disc to DVD movie and hit 'start' to play.
Keep in mind that you'lll want to have a modchip that supports Dual Layer DVDs so those of you with homebrew chips will want to use the recently released YAOSM 1.8 firmware.
The application can be download from Team Symbiote Wii DVD Player Beta 1.0