Chekhov's Exhibit

Some fancy object is sitting out in the open at a museum, carefully protected by velvet ropes and laser-security systems. Sometimes it's encased in a carefully monitored glass case. Fat chance it'll stay there for the entire movie.

In caper movies, objects like this are just asking to be stolen via a "Mission Impossible" Cable Drop. Rest assured, if a tour group is looking at it in Act One, expect it to be stolen by Act Three. In horror movies, the object will be Sealed Evil in a Can. In action films, nearly anything is possible: the object might need stealing, or it might be a bomb... it all depends on the plot. Often overlaps with Break Out the Museum Piece.

Examples of Chekhov's Exhibit include:

Fan Works

With Strings Attached: The first piece of the Vasyn is in New Zork's Museum of Natural History. Since no one there has any idea that the thing has value to anyone besides geologists, or that anyone could possibly steal it—it's 20 feet of solid rock, weighing at least 90 tons—it's not guarded at all.

Film

Superman Returns includes an early shot of a kryptonite shard sitting under glass at the Metropolis Museum of Science. Naturally, Lex Luthor shows up eventually to steal the thing.
The Declaration of Independence in National Treasure.
A lot of people forget that the Pink Panther was originally a diamond targeted by thieves, and not the cartoon cat. It was kept in a glass case, naturally.
The Etoile de Joie diamond in Herbie Goes to Monte Carlo is on display in a museum.
The DaVinci Journals in Hudson Hawk.
In the film Sky High Steve Stronghold makes a point of showing Royal Pain's pacifier to his son during Will's first trip to the secret sanctum. Of course, its theft is important to the plot.
Sammael is mystically imprisoned inside a sepulcher, on display at the Machen Library of Paranormal Artifacts in the movie version of Hellboy.
Moonraker featured an entire museum full of glass artifacts. Not thirty minutes later, we get to see every last bit of it trashed when Bond fights The Dragon, a Kendo champion who insists on going after him with a shinai. Bond manages to repel him with a glass-handled sword that was shown earlier.
In the film The Phantom one of the Skulls of Touganda that the bad guys (and the good guys, for that matter) need can be found at a museum exhibit. One that both the bad guys and good guys happen to visit at the same time.
Get Smart has the shoe-phone and the car, although the car is a subversion.
The Chinese mask in Entrapment.
Ghostbusters 2: A painting of the tyrant Vigo at the Metropolitan Museum of Art sets off the chain reaction that fuels the plot.

Literature

A boy called Kevin in Matthew Reilly's Area 7. He's in a carefully monitored glass case, that is in the military's version of out in the open - that is in a very open position in a secret underground labratory. The President goes on a tour of said labratory, including the case, at the start of the book. That boy is not going to stay in his case for long.
In Against a Dark Background the Crownstar Addendum is the most valuable piece of jewelry in the solar system. In an aversion of this trope, it's kept in a glass case... in a heavily-secured vault guarded by the entire military force of a city-state. And in a further subversion, the thing in the glass case isn't actually the real Crownstar Addendum: the genuine article is kept hidden behind a random metal panel wrapped in oily rags.
The Panarch's museum in the second The Wheel of Time book contains a few ancient magical artifacts along with mammoth bones and other random junk. One of the magical artifacts happens to be one of seven seals binding the Dark One's prison closed.
In the Discworld book Men at Arms, the Assassins lock up the Gonne, it gets stolen, and the rest of the plot follows from there.
Dresden Files: the dinosaur skeleton in the museum in Dead Beat becomes important later. ZOMBIE DINOSAUR.

Live-Action TV

The Hans Christian Anderson manuscript in the Hustle episode "Law and Corruption".
The Highlander episode The Lady And The Tiger featured a McGuffin under glass that the immortal cat burglar Amanda Devereaux wanted to steal before her former-partner-turned nemesis Zachary Blaine (played by Jason Isaacs, very early in his career) got to them.
- Another episode saw MacLeod asking for Amanda's assistance in retrieving the Cross of Saint Antoine from the house of the Evil Immortal of the Week. The EIOTW kept the cross in a big glass case in his foyer.
Sherlock, "The Great Game": When Watson sees the news report about the Baker Street explosion, there's also a news item about the new Vermeer painting at the museum. It doesn't get stolen, but it does turn out to be important.
The pilot Miniseries of Battlestar Galactica prominently featured a recently-set-up museum exhibit in one of Galactica‍'‍s hangar pods, centered around a lineup of Viper Mk IIs. The mechanics end up refitting them for action after their entire complement of modern fighters is wiped out in the opening Cylon attacks.
In the Doctor Who episode "Planet of the Dead", Lady Christine de Souza steals the solid gold cup of King Aethelstan from a museum by way of a carefully timed "Mission Impossible" Cable Drop while the heavily armed guards are still in the room.
- It probably helped that they didn't even bother putting the cup in a case - it was just sitting out in the open.
On the season one finale of Leverage, this is subverted as they steal the rest of the art surrounded the Two Davids exhibit and leave that exhibit alone. Their motive was to cause the CEO of the insurance company to be fired and the rest of the art in the exhibit was insured by his company as opposed to the Two Davids exhibit which he owned personally.

Video Games

Happens in the introduction of Wario Land The Shake Dimension, with the globe containing the whole mini universe seen in a museum, stolen by Captain Syrup and sent to Wario (So that she could get Wario to enter, beat the Shake King and bring the Bottomless Coin Purse to the outside world.)
The Expel Mirror in Shin Megami Tensei: Persona is introduced as the centerpiece of a museum exhibit. It turns out later that you need the mirror in order to fight a boss; however, the only obstacle to the protagonists finding it is that going to the museum in the first place is entirely optional, so many players will have no idea where to look for it.
In Ultima IX the Avatar Museum held dozens of past games' MacGuffins under glass, and seemed the obvious repository for puzzle-solving implements to be taken by the player or seized as weapons by the Big Bad. Inverted and subverted; the heist already took place before the player arrived at the scene, and none of the rest of the exhibit becomes relevant again.
- In Ultima VII, all eight of the original moonstones are sitting unsecured in a museum in Britain, waiting for the Avatar to take them.
The golems on the exhibit grounds in Wild ARMs 1. Inverted that you're going to fight them later in the game.
The seemingly useless stone in the museum in Pokémon Black and White. It is actually a legendary Pokémon, which you must catch to advance the plot.
Ghostbusters the Video Game: An exhibit at the Natural History museum regarding Gozer worship triggers a shockwave that re-awakens all kinds of booby traps and leftover followers Ivo Shandor left behind.

Web Original

In one Global Guardians story, the Unaussprechlichen Kulten, a book of black magic rituals, was put on display as part of a special "Magic and the Occult" exhibit at the Field Museum in Chicago. Naturally, it was stolen.
In the Whateley Universe some people are Genre Savvy enough to think about this. After Team Kimba stops a team of mutant ninjas raiding the campus, the school has the secret devise the ninjas were carrying, and the headmistress shows them the glass case in the school museum. But it's a hologram of the real thing, which is locked up a lot more securely.
- Still, they pull this trope out at Halloween when the headmistress is fighting Deathlist.

Western Animation

A few of the Talismans in the Jackie Chan Adventures were like this.
Happens in The Real Ghostbusters, notably in a Lovecraft tribute episode where, once again, the Necronomicon was on display to the public. You'd think they'd learn after a while.