So in Zelda II: Adventure of Link, you get a game over if you die. Happens all the time, I know.
BUT not only do you die, you fail at staying alive. They go hand in hand, normally, but in this game there is a consequence to dying that is unrelated to the actual Adventure of Link.

See, Link is adventuring to save Zelda (actually I not II), a sleeping princess.

And if you fail, by death, your corpse is harvested for blood by Ganon's minions to resurrect Ganon from Link's previous adventure of saving Zelda (II or unrelated? we're just not sure).

That's just a consequence, not really related to your game. And the death doesn't even have to be a killing by one of Ganon's minions, it can be a death for any reason.

Now I have to ask, what happens if Link dies from old age? Does Ganon still get resurrected?

Not only your death, but the consequences of your failure, which are actually unrelated to what your quest was.