What Happens After A Zelda "Game Over" Screen?

Although Link's Awakening occupies a place in the middle of one of the official Zelda timeline's branches, what you find in the Face Shrine suggests otherwise. It's a small detail, sure, but the inscription on the wall suggests that Link's dead. Or at the least, somehow in or on the Wind Fish.

The inscription reads, in full: 

“TO THE FINDER… THE ISLE OF KOHOLINT , IS BUT AN ILLUSION… HUMAN, MONSTER, SEA, SKY… A SCENE ON THE LID OF A SLEEPER’S EYE… AWAKE THE DREAMER, AND KOHOLINT WILL VANISH MUCH LIKE A BUBBLE ON A NEEDLE… CASTAWAY, YOU SHOULD KNOW THE TRUTH!”
(As quoted in Zelda Dungeon's walkthrough) 

This inscription confirms that Koholint island is just an illusion. In fact, it goes so far as to imply that it's just the Wind Fish's dream. Just to the North of the Face Shrine, though, is an owl statue. 

In Link's Awakening these statues work kind of like the Gossip Stones in Ocarina of Time and Majora's Mask. They share in-world information, although the owl statues are a fair bit more cryptic. Especially the one in question, which says: 

“THE WIND FISH SLUMBERS LONG… THE HERO’S LIFE GONE…”

Definitely making a connection between the illusory island and Link's being dead. Though it's ambiguous about whether Link's dead only while the Wind Fish sleeps, or if Link is dead and the Wind Fish's dream is some kind of afterlife. 

Since most of the games are supposed to star a different Link and Zelda, the hero's being dead isn't out of the question, but it would be absolutely wild to know for sure.