From the Opera forums...

The Opera browser on Wii does support playback of movies on Youtube. Youtube seems to have broken something tonight, however, blocking Wii users. We will investigate and contact them about it.

--
Wilhelm
Opera QA (Wii)

Well I am glad to hear that the functionality wasn't pulled for some reason. Hopefully they will have things up and running sometime soon! Thanks to VillyVombat for the heads up.