I wish that my answer were more obscure and esoteric, but I'm honestly going to be with the popular opinion and vote for Link and Zelda! Their's is a tale of high fantasy and fate that spans generations of Links and Zeldas! Moreover, by the time we reach Ocarina of Time: Zelda's character has been expanded well beyond the damsel in distress cliché! (For very similar reasons, I would vote for Ganon(dorf) as one of the greatest antagonists in video games.)
I consider myself to be a very liberal and progressive person and am totally for expanding this list to consider couples beyond the traditional heterosexual pairings. In fact, it is worth noting that Birdo's gender and orientation is rather ambiguous, unspecified, and fluid--with some games suggesting an effeminate female, others suggesting an effeminate cross-dressing homosexual male, and still others suggesting a transgendered male-to-female! In addition to that, Yoshi and Birdo are clearly not even the same species! However, I cringed somewhat at the Mario and Luigi post--they're unquestionably biological brothers! Although the risk of breeding genetically deficient children doesn't apply to homosexual incest, it's still incest and is a very icky proposition! (The same goes for all of the slash fiction out there about Supernatural's Winchester brothers, Thor and Loki, etc!)