Teleport Gun

Gun-like technological devices, frequently designed with this purpose in mind, or regular old Ray Guns combined with a teleporter to fake a death, are a common variant: The Teleport Gun.

This is basically the Impossibly Cool Weapon crossed with Teleporters and Transporters.

It's a Ray Gun or Energy Weapon-like device that usually resembles a gun or rifle, with a trigger, barrel, grip, etc., whose purpose is (usually) the teleportation of its target (or sometimes, the teleportation of its user), often but not always instantaneously. There are exceptions, such as weapons that teleport their ammunition rather than their target, or which deploy a portable teleportation system for the user; there are also examples of Teleport Guns disguised as ordinary objects.

Another significant variant is when the use of regular old Energy Weapon and a teleportation device are combined to fake a person's death by teleporting the target an instant before they would be hit by the weapon discharge, as the teleportation effect is similar to the in-universe appearance of molecular disintegration from an Energy Weapon. To observers, the subject will appear to have been vaporized.

If the target is not teleported, but rather the user, than it is usually for escape or a tactical advantage.

The Teleport Gun is frequently an invention of the Mad Scientist.

Inversions and other Subversions, like guns that teleport bullets through walls to reach a target, or guns that teleport virtually endless ammunition from another location into their magazines, are worthy of mention. Subtrope of Weaponized Teleportation.

Examples of Teleport Gun include:

Anime and Manga

There is a non-lethal example in Mahou Sensei Negima. During the Mahora Festival battle, the Big Bad of the arc is actually a Well-Intentioned Extremist who uses exclusively Non-Lethal Warfare and, on the other hand, her goal is not to disable the opponents permanently, but only for the next few hours. To achieve this, she starts using using special Magitek bullets which send whatever they hit three hours into the future.
In Gantz, the "Y-Gun" traps its victims with a net, then teleports them to an unknown location.

Card Games

The Star Munchkin card game has a card titled "Foof Gun". It gives a big bonus against enemies, but using it forfeits going up levels, since you don't kill them but teleport them away.

Live Action Television

Star Trek: The Next Generation: "Gambit," Part 1- Picard is supposedly vaporized by an Energy Weapon beam in a bar fight. Turns out he was transported to a mercenary ship.
Star Trek: Deep Space Nine: "Inter Arma Silent Leges"- We see a Romulan disruptor used on someone. Next scene, they're still alive. Turns out that what we thought was the disruptor beam was actually a transporter.
- Actually, the weapon was referred to as a Phaser, not a Disruptor (different mechanics- a Phaser is more versatile, with multiple non-lethal settings). That being established, it's possible the weapon was an integral part of the illusion: it may have been modified into a "true" Teleport Gun rather than simply relying on beaming the target away a split second before a Phaser set on "kill" or a disruptor (without a "stun" setting) could vaporize him.
- Another episode of Star Trek: Deep Space Nine featured a projectile rifle that was modified with a special scope and a microtransporter, enabling the user to fire bullets through walls, bulkheads and (presumably, though not confirmed) force fields. Not a straight example (since the ammunition was transported, not the target or user), but played with enough to warrant mention.
The Timer in Sliders was more like a remote control than a gun, but it was occasionally used for this exact function on villains of the week, by the main characters opening up a surprise wormhole that sucked the villain away into another dimension, or possibly oblivion. Logan St. Clair met her fate this way.
Several episodes of Earth: Final Conflict featured a black market device based on Taelon technology that was like a harpoon gun that deployed an interdimensional Gate for personal use.
Time Trax: Darian Lambert is a cop from the future who has come back in time to retreive criminals who escaped from 2193. When he finds them he shoots them with a beam or a pellet gun disguised as a car alarm remote; the pellet or beam (Depending on the Writer) doses them with some Phlebotinum and sends a signal that the person is ready for transport.
SeaQuest DSV: The holographic programmed defender of the alien ship encountered by the submarine crew in Season 1 had a bracelet-like weapon that dematerialized people into thin air. They could be rematerialized, if the user chose to do so.

Tabletop RPG

Champions campaign The Great Super Villain Contest. A section giving advice on using sub-plots mentions a "Teleportation Blaster" that could send its target to another location. The specific example involved sending the target to another dimension.
Warhammer 40,000: the Ork's Shokk Attack Gun works by teleporting snotlings through the Warp and into vehicles and Power Armor. Their short trip through the warp makes them go completely insane, so when they arrive they swarm anything they can, essentially paralyzing the unit.
Rogue Trader makes mention of teleporting a few virus grenades onto the bridge of an enemy vessel.
GURPS has Affliction with the Advantage enhancement, which can be used to afflict the Warp advantage, which teleports the target. (It gives this as an example.)
Champions: Teleportation can be bought with the advantage "Usable As Attack", a subset of "Usable on Others". This allows the user to teleport a victim somewhere else. If the victim is teleported into a solid object, they will take damage.

Videogames

In Chex Quest the player uses a variety of "Zorch" weapons that are in fact gun-like teleportation devices. They are used to zap the enemy Flemoids back to their home dimension. (In other words, teleportation is used as a substitute for killing in this kid-friendly game.) Full teleportation takes multiple shots, unlike most examples.
The Portal Gun in Portal, though the mechanics are slightly different: the gun shoots connected orange and blue portals but does not instantaneously teleport its target.
This is an Inverted Trope in the game Xenosaga as most of KOS-MOS projectile weaponry, transports its ammo to the gun from huge stockpiles which is then fired.
The Displacer from Half Life: Opposing Force. It teleports its victims to the dimension Xen, and you can also use it on yourself in order to hop into Xen and find some resources there (as well as a convenient portal back to Earth.)
Viki is a Recurring Character in the Suikoden series, whose entire skill set is based on her Blink Rune, which allows her to teleport people and objects across great distances. From Suikoden II, onward, she can be used as an active party member. In battle, she either: a) teleports random objects (including herself!) onto enemies heads, b) randomly teleports an enemy off the field, or c) randomly teleports a party member off the field. The last one is especially worrisome since there's a slim possibility she might teleport the entire party away, leaving only herself!!
Words Worth: Maria may be a homage to Viki, since she dresses in white, has blue hair, and has teleportation as her main offensive weapon. In both the game, and the anime adaptation, she blasts Astral twenty years into the future with her Mystral Window ability!
Unreal Tournament 3 has a gun whose primary fire is a beacon and secondary fire teleports the user to the beacon. Hitting an enemy with the beacon allows for a Tele Frag.

Western Animation

Used in an episode of The Fairly OddParents, where it was used to fake Mark Chang's death so he could escape from his Yandere alien princess bride.
One episode of Futurama featured a Teleport Gun that everybody thought to be a disintegrator.