Shadow Link's death. He knew that by sacrificing himself for a world that would never appreciate what he'd done, would never care for a "shadow" like him, he was casting away his life for a bunch of ungrateful and unknowing people who would celebrate the four Heroes, and yet he still did it because he wanted to show them he was a hero, too; that not all dark things were bad. The most heart-breaking part of this was when the four Links are gathered around the dying Shadow, and they talk about him. Violet even refers to them, the four heroes of light, as Shadow's family, despite him being what he was and what he did.
When Link's father found his son unconscious in the castle. It's Shadow Link.
Skyward Sword Manga
The death of the first Link. Although Skyloft was successfully lifted away to the sky, Link, who suffered betrayal and imprisonment for years, is mortally wounded and chooses not to join his comrades. As Link lays dying on the surface, he admits that he wanted to go up to the sky and fly with the Loftwings, but his wounds made it impossible for him to do so. The scene becomes even more heartbreaking when Goddess Hylia discovers the recently-deceased Link lying nearby and mourns for her hero's passing, revealing that Link's imprisonment was intentional as a means to turn him into her chosen hero. Regretting her actions, she swears to Link that his spirit shall be reborn while she shall shed her divinity so that they shall meet each other again as simple humans.
Ocarina of Time Manga
When Link has to kill Volvagia even though it was his childhood pet that he bought before drawing the Master Sword. This, along with the cuteness of Volvagia in Link's childhood, finishing with Volvagia's dying words being "Link", it's enough to make an adult man cry.
The Game Over screens in any of the 3D games. Seeing Link fall down while depressing music plays in the background and the words "Game Over" fade into the screen. Spirit Tracks' game over screen was the worst, for this troper, because the overworld theme he adored was suddenly being slowed down and symbolizing your failure, rather than your adventure throughout the land.
Also from Spirit Tracks, getting a Game Over while on the train needs special mention. Think about it: The train is blowing up, Link is (probably) laying unconscious, and Spirit-Zelda is is shrieking "AAAAAAAHH! LINK!" at the top of her distraught, grief-filled voice. Even though the game has no real voice acting, Zelda's shriek still echoes on the mind.
Things get even worse when you have Teacher with you, who is so frantically worrying over Zelda and searching for her endlessly when you could assure him that he's fretting too much, but won't. And things get even worse than that if you crash while he's on the train. During the resulting explosion, he'll state a final line in which he can actually see Princess Zelda. Who happens to be a ghost. Oh, dear, indeed.
The Zelda II manga by Ran Maru runs into this quite a few times. It's bad enough that the first game's Zelda is crucified and killed off midway through the story, but later on the body reappears in a brief Hope Spot, only to be dissolved in a lava pit shortly after.
On a slightly less tragic note, a large monster Link befriends suffers a similar fate after performing a Heroic Sacrifice. When Link looks over the cliff, only its burning ribbon remains. However, they reappear at the very end, with the implication that the Triforce was involved.
The cartoon has an episode where Ganon captures Link when he was aiming for Zelda, leaving the princess distraught. Worse is that only Link's body was captured, leaving his spirit to wander as a ghost until Zelda can reunite them.
The timeline in general can be this for certain games; specifically those that involve another world entirely like Majora's Mask and A Link Between Worlds. You may have saved Termina and Lorule in one timeline, but their fate in the other timelines is left entirely ambiguous due to there being no game in the other timelines involving those places. Who's going to stop the moon from crashing in the decline timeline? What's become of Lorule without its Triforce in the child timeline? Until we get a definitive answer, it's likely to assume the worst for these two...