No "Off" Button

Perhaps it's some sort of Doomsday Device or an Artifact of Doom maybe it runs on some sort of perpetual energy. Possibly there are actually two phlebotinum devices interacting with each other and destroying the world, but if you turn off one device the other one will grow strong enough to kill everyone on earth anyway. Perhaps it's a Clingy MacGuffin or Clingy Costume and it's draining your life force, but if you remove it you'll blow up. No "Off" Button is simply the reason a particular piece of dangerous Applied Phlebotinum can't be turned off or destroyed, such as it lacking an off button.

Obviously this applies to any kind of Doomsday Device, since turning it off would defeat its purpose.

Failsafe Failure is closely related, and involves examples where there is an "off button" or other form of Override Command in place, but it doesn't work.

Examples of No "Off" Button include:

Anime and Manga

In Naruto, the Akimichi pepper pill puts your metabolism into overdrive, converting calories into tremendous power. But it keeps going even when your body has no fat left to use up.

Film

Star Trek II: The Wrath of Khan, David Marcus helped build the Genesis Device.

David: It's the Genesis Wave! They're on a build-up to detonation! [snip]
Kirk: We'll beam aboard and stop it.
David: You can't.

Forbidden Planet.

Morpheus: Son, turn that disc. The switch. Throw it. In 24 hours, you must be 100 million miles out in space. The Krell furnaces...chain reaction...they cannot be reversed.

The U.S. and Soviet supercomputers in the film Colossus: The Forbin Project.
The Doomsday Device in Dr. Strangelove will automatically detonate if anyone attempts to shut it off.
- In this case, the designers of the device built it like this intentionally, and to them it made sense. It's meant as the ultimate, permanent deterrent to nuclear warfare; if there were any way to stop it once it's turned on, it wouldn't be able to carry out its function. It wouldn't have needed an off switch if not for a few miscommunications.
In Highlander 2: The Quickening the planetary shield couldn't be taken down even though blocking off the sunlight was killing the planet, the inability to do so being attributed to a lack of energy. In the end General Katana is killed in one of the shield's beams, and his Quickening provides the energy needed to take it down. At least, if one is being charitable, one can assume that's what was supposed to be happening there.

Literature

In Robert Sheckley's short story The Laxian Key, a pair of ne'er-do-wells find a machine that produces some substance in unlimited quantities but can only be turned off with the eponymous "Key". When the substance begins to flood their ship, they try to sell the machine to an alien race who feeds on said substance, but are almost blown to pieces by their navy. It turns out, these aliens already bought several such devices, which completely ruined their homeworld, so they tell the heroes to come back with the "Laxian Key" and ask any price for it.

Live-Action TV

Happens pretty much every week on Eureka. A big piece of Tech is endangering the town, and Jack Carter asks something related to the easiest way to shut it off ("Why don't you just unplug it?"). Oh guess what, if you turn the device off, the town'll catch on fire, or something along those lines.
- This is a town where a boiler in the house is more likely to be a nuclear reactor, requiring a Ph.D. in Nuclear Physics to fix.
On one episode of Buffy the Vampire Slayer, there was a lightning wielding gauntlet that could only be destroyed through a specific ritual.
On the various Star Trek incarnations, Holodecks have some kind of independent power source, so whenever they go haywire and stop responding to commands, they usually can't simply pull the plug.
Happens a couple of times on Charmed, such as when Billie put on Hippolya's Belt and became a superheroine. The belt was killing her and the only way to remove it was through a specific spell found only in Magic School's library.
In Stargate SG-1 and Stargate Atlantis, the body-swapping device didn't have an off-switch (or at least it was never discovered) so every time someone activated it, he had to destroy it.

Music

A certain Bawdy Song about a sex machine with a Bloody Great Wheel includes the ominous rhyme, "And now we come to the tragic bit, there was no way of stopping it".

Radio

In The Hitchhikers Guide to The Galaxy radio series, there's a malfunctioning clone machine which keeps starting new clones when the previous clone is only partly done. Its operators can't turn it off because if they did, they'd be liable for murdering the clones that are in mid-cloning when the button got pushed.

Video Games

In Zak McKracken and the Alien Mindbenders, the titular Mindbender only have a big ON/ON switch; throwing it seemingly does something (reverses the polarity?), but turning it off is impossible. Given that it's designed to induce stupidity, this is a perfectly sensible design choice. Fortunately, it can be destroyed by the Skolarian Device.

Western Animation

In a mundane version, once on The Simpsons when Homer decided to become an inventor:

Homer: Now, here's my "Everything's O.K." alarm!
Homer flips a switch the device, and it begins to emit a high pitched, incredibly loud beep. The rest of the Simpsons cover their ears as Homer speaks up.
Homer: This will sound every three seconds, unless something isn't okay!
Marge: Turn it off, Homer!
Homer: It can't be turned off! [alarm fizzles out] But it, uh, does break easily.

- Another Simpsons example; in the episode where a shady dealer sells Springfield a monorail, and it ends up going out of control, cutting the power is suggested, but the monorail is run completely on solar power... the monorail stops only when an eclipse happens... and then when the eclipse ends, the monorail keeps right on going.
An episode of Taz-Mania guest-starred Marvin the Martian who sets up a device to destroy the Earth rigging it, so once it's activated, even he can't shut it off. After turning it on, he realizes he's made a slight miscalculation:

Marvin: In 10 minutes it will all be over for you pitiful Earthlings. My ship will be here to pick me up in approximately one hour so.... Uh oh.

Real Life

Certain devices that rely on chemical reactions obviously can't be easily shut down once the reaction starts, unless you have control over the access to the reactants themselves, such as some types of rockets and missiles (IIRC,^{[please verify]} more of a fundamental quality of solid-fuel rockets).