Faux Death

When William Shakespeare put Juliet into a death-like state in Romeo and Juliet, he was probably using a trope that was already lying around. He may have grabbed it from an old Celtic precursor of Sleeping Beauty.

Anyway, this one has some years on it.

In this trope, people who are not dead appear to be dead and, like Human Popsicles, do not age. Because they look like that, all kinds of ugly stuff happens, either to them or to the ones they love, up to and including being kissed by princes.

When intentional, it's a kind of Faking the Dead. If someone is merely sleeping the sleep of the dead, but life can be discerned, see Deep Sleep.

Supertrope of King in the Mountain.

As a Death Trope, Spoilers ahead may be unmarked. Beware.

Examples of Faux Death include:

Anime and Manga

Zabuza Momochi in Naruto. Also Faking the Dead, because Haku used this trope to protect Zabuza from Kakashi and his squad.
Griffith of Berserk blackmails Foss, the leader of the conspiracy to kill him, into placing a drug that does this into his goblet instead of the poison the conspiracy intended to be placed as part of his masterful Batman Gambit that ultimately leads to the Queen and her nobles being locked inside a burning castle to die.
After Wolfram's heart is stopped in Kyou Kara Maou, his body is put in a nice little magical life-support box until his fiancé, Yuuri, can defeat the Big Bad and get him going again. It's kind of up to interpretation if he was actually at any point dead or not, but Yuuri certainly has a strong opinion that he wasn't.

Comic Book

The origin story of The Spirit.

Film

Happens to poor Frodo in The Lord of the Rings: Return of the King (or The Two Towers, if you go by the literature) when Shelob stings him. His death-like state is convincing enough for Sam, at least, until the point when the Orcs take Frodo and explain that he's still alive in Sam's earshot, at which point Sam decides to follow them to the orcs' barracks to get his Big Damn Heroes on.
In Superman Returns Everyone laments Superman's apparent death. But when Lois and Jason go to see him, there a (very clear) faint heartbeat on the monitor. He hasn't even died!
- It's visible as Jason and Lois leave the Daily Planet that Perry White has two templates of the front page prepared ahead of time: one announcing Superman's death and the other announcing his recovery.

White: Always be prepared.

Folklore

Obviously, any version of "Sleeping Beauty" (see here for a list of many), or "Snow White" (see here). True Love's Kiss, stat!
- The Pentamerone has a Gender Flip variant as its Framing Device.
King Arthur is sleeping until his return—hence "The Once and Future King".

Literature

Deconstructed in Neil Gaiman's Snow, Glass, Apples, a Grimmified Perspective Flip where Snow White is a bloodthirsty vampire, the evil queen is a benevolent ruler who put Snow White into a deep sleep to protect the populace, and the prince who accidentally woke her is a necrophiliac.
The Count of Monte Cristo ? in fact, that entire romantic subplot between Maximilian and Valentine was a rather obvious Shout-Out to Romeo and Juliet. Except it had a happy ending... sort of.
Happens to poor Frodo in The Lord of the Rings: The Two Towers (or Return of the King in the Jackson-films) when Shelob stings him. His death-like state is convincing enough for Sam, at least, until the point when the Orcs take Frodo and explain that he's still alive in Sam's earshot, at which point Sam decides to follow them to get his Big Damn Heroes on.
In Graham McNeill's Warhammer 40,000 Ultramarines novels, Roboute Guilliman is seen in his stasis tomb, and we are told of legends that he is healing from his wound and will arise again.
A state that Granny Weatherwax enters in Discworld whenever she's Borrowing. It had caused enough unnecessary embarrassments that she now wears a small cardboard sign with the words "I Aten't Dead".
Little, Big draws on the legends that German emperor Friedrich Barbarossa is asleep under a mountain by having him wake up.
In Edgar Rice Burroughs's The Master Mind of Mars, Ras Thavas does this to preserve the bodies he swaps (or swaps parts of). When Valla Dia is in danger, Ulysses Paxton resorts to it as the only way to hide her safely.
- In the Disney movie John uses puffer fish tetrodotoxin to lure a Thern spy out of hiding so he can steal his medallion and use it to return to Mars.
In Sir Arthur Conan Doyle's The Poison Belt, people start dropping like flies as the Earth passes through the poison belt. Professor Challenger uses bottled oxygen to keep himself and the other protagonists conscious for a few hours so that they can observe the death of humanity before joining it in death; they're all very surprised to wake up and find that they're alive. And even more surprised to find, some twelve hours later, that everyone else wakes up too! (Well, except for the ones who'd been killed in the accidents and fires that occurred when everyone first passed out.)
In Robert E. Howard's "The Devil in Iron" Conan the Barbarian finds a castle appearing where he knew there had been a ruin, and inside, a woman who has just waking up, thinking that historical events were just last night.
- In "Jewels of Gwahlur", Yelaya.
The immortals of The Madness Season have a technique that allows them to impersonate a walking corpse. The most skilled ones are able to temporarily stop their metabolic functions. The hero's father ultimately died when he went too far in his corpse transformation. Daetrin himself almost wound up doing the same.

Live Action TV

Lost: Paralyzed by spider bites into a death-like state, a couple are buried alive. They died rich, though, so it wasn't a complete downer.
Jack Bauer in the season 4 finale of 24, when he found it necessary to fake his own death in order to prevent being Killed Off for Real. Coupled with his being clinically dead for several minutes during season 2, it inspired the Jack Bauer Fact "Jack Bauer died for his country and lived to tell about it. Twice."
River and Simon do this in Firefly, to get into a hospital. Later, Tracy uses it as a way to run away.
Happens at least once to Sloane on Alias, which also featured many other examples of Faking the Dead.
On one episode of The Master (a.k.a. Mystery Science Theater 3000 stalwart Master Ninja), McNinja master McAllister (Lee Van Cleef) reveals that he can accomplish this by meditation; this turns out to be an example of Chekhov's Skill.
There's a grand tradition of this in Xena: Warrior Princess, dating all the way back to Xena's "death" by poisoning towards the end of the first season. (Since the trope was still fresh at the time, the resulting mourning process among her friends is thoroughly affecting.)
Tek Wars tried fooling an AI with the Human Popsicle trick, because it detected the Cryo virus to still be active. Said AI figured they were finally killed by the stronger-than-normal setting of the Cyro tube.
It is rather unclear whether Babylon 5's Captain John J. Sheridan really died on Za'Ha'Dum or if Lorien kept him suspended in a state near death (between tick and tock). Lorien says "He was dying, he was dead" but he has a tendency to be vague. Zack Allen doesn't know anymore, and Michael Garibaldi is skeptical. In either case:

Drazi Ambassador: We thought you were dead.
Captain Sheridan: I was. I'm better now.
and later
Captain Sheridan: Death! Been there. Done that.

Cameron gets one in the Stargate SG-1 episode "Babylon", in order to get out of a revenge duel to the death.
Captain Kirk, from "Star Trek: The Original Series", had a rather 'Romeo and Juliet'-esque faux death in "Amok Time", that was caused by the Doctor...of all people.
On Heroes, Sylar tricks the Company's doctors into removing his restraints by stopping all of his vital signs.
Inverted in legendary fashion in the Parrot Sketch.
Fraser did this on Due South with an Inuit concoction that slowed his body down.
Arthur did it on Merlin to lure his father into crying the tears of true remorse needed to break the troll magic spell. He did, however, require an antidote to stop the potion he took killing him for real.

Music

The Irish folk song "Finnegan's Wake".

Theater

Juliet, of course. She woke up to a dead boyfriend. That she liked. Bummer.
- Husband, by that time. For one night or so. Ouch. There goes the honeymoon.

Video Games

Naked Snake in MGS3: Snake Eater has a pill that can induce this state to fool enemies into believing him dead. If you let him go too far before using the revival pill, however, he really will die.
King's Quest VI Heir Today Gone Tomorrow allows protagonist Alexander to fake his death with a potion that actually does kill him, but then wears off after a few minutes. The in-game purpose of this is to fool a spy into reporting to the Big Bad Vizier in order to lower the security at the castle you inevitably have to infiltrate, but the actual purpose (because the security is the same either way) is to trigger a cutscene in which you can see the genie's lamp, so that you can replace it with a fake one later on.
Hitman: Blood Money. A special drug used originally in mission 3, to get a target out of a rehab clinic without actually killing him. The second time it is used is on 47 himself, courtesy of Diana. Admittedly, to she does this to save his life.
The Elder Scrolls 4: Oblivion features this as part of an Assassin's Guild quest - the player needs to cut the target with a poisoned dagger to fake his death in front of someone who is trying to kill him for real.
Guybrush Threepwood pulls this off in The Curse of Monkey Island with a combination of alcohol and a homemade hangover cure. Twice.
In Team Fortress 2, the Spy can pull this off regularly with the use of his Dead Ringer watch.

Web Comics

Tessa of SSDD was once mistaken for dead after being shot with a tranquilizer-coated bullet due to her artificial heart not having a pulse.
In Roomies Codrus tried to get his former superiors in the Fox Empire off his back by faking his death with a potion. Unfortunately it temporarily turned him into a zombie when he revived early.

Real Life

Truth in Television: The right amount of puffer fish toxins can cause a person to appear to be dead to even doctors. There have been several cases where people have been declared dead and returned, especially in Jamaica, where this method factors heavily into local Zombie Lore.
- Same with the toxin of an Australian blue-ringed octopus, which paralyzes the victim and stops their breathing (and sometimes causes temporary blindness and deafness as well) but doesn't kill immediately and does wear off eventually. If somebody has the sense to perform rescue breathing until the paramedics arrive with mechanical assistance, the victim can make a complete recovery. If not...
- Fun fact: the toxin of puffer fish and blue-ring octopus is one and the same; both groups of creatures obtain the substance through commensal bacteria.
Even after the heart stops beating and the lungs stop breathing, the brain is still functional for a few minutes. That's why CPR sometimes works -- not often, well below 5% according to some assessments, but there's still a chance if you try.
A person can survive 40 minutes or longer in freezing water, because everything slows down. They appear quite dead and frozen, but if they can be pulled out,oxygen gotten to the brain, and the effects of hypothermia counteracted, they can still be revived sometimes.