To Blue Reptile and Cocco002 ...
I really don't think we should buy too much in to the speculation that they are dead. The closest evidence we have is in Twilight Princess. I'll get to that in a moment but I wanna say for now that we should not hold TP to change our theories for ALL games.
In other games, to me, it seems pretty strait forward. I'm talking about Ocarina of Time and the Wind Waker. It makes sense that in OoT, they are simple spirited away to the light realm to become Sages. The light realm is their sacred temple and since they have responsibilities (like Zelda did in SS -- she couldn't just run and talk with Link even for a moment), Sages such as Saria, for example, cannot return home. This is why people are mourning for her. This is why she leaves without a trace. Once she is awakened as a sage she runs off to fulfill her duties.
The reason we see people who seem like ghosts, is because the Sages have magical powers. Heres my idea -- I think they can telepathically project themselves outside the light realm (or wherever the sage is supposed to be, depending on the game). I think they do it on special circumstances. Rauru needed to help Link on his quest so he showed up on earth (or, Hyrule rather) as the physical form of Kaepora Gaebora. He was kind of like a Prophet sent from heaven in a physical form (Sorry, I use a religious references here to help illustrate my point, and because the producers have used religious parallels for inspiration in the past.). And as for the reincarnation of Rauru, I think BlueWindow summed that theory up ^above^. The king of the red lions in WW needed to stay in the temple, so he transferred his spirit into something physical too - the boat- in order to help the hero with his quest. I think the Sages go awol in physical shapes when the hero looks like he's in trouble or looks like he is in danger of failing his quest to conquer evil. And, apparently, to run to death mountain and celebrate lol.
Now that I mention it, I think the reason they were on death mountain and not lon lon should not be too looked into. I've seen those credits and people are kind of celebrating everywhere. I think the reason they are off to the side, on a mountain top, is simply to reinforce that they are now otherworldly beings. They are sages. They are on a different plane than Hyrulians now.
As for the TP sages. I dunno... TP includes the Dark Tribe. The Dark Tribe/ Elder Sages , the Twili, The alternate dimention and the Dark Realm all kind of throw a crazy wrench into things. considering you have many more visual hints to the sages being alive as compared to just TP showing seemingly dead sages, I would just go woth the more common idea. I think TP was weird. Not superficially, but rather as compared to the plots of other games. It kinda threw me off. We don't even know what the sages were doing. Maybe they were operating telepathically from the light realm when they attempted to execute ganon. Haha I have no idea. But it is such a confusing part of the story line that I try not to let it change my thoughts of other story lines.