Seen It a Million Times.
Basically, a machine or creature is programmed to perform specific kinds of duties, but at some point of time something happens to it's environment that makes continuing to perform the task pointless. The machine, ever faithful, keeps doing it anyway since it was never told to stop. Often, this is a result of After the End or at least all humans leaving the place and forgetting the machine.
It can also happen to a person if this person is somehow traumatized and has becomes a stoic being with no sense of it's surroundings.
See also Bothering by the Book and The Determinator.
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At the end of an EC Comics story a man's Robot Wife keeps protecting him long after he's dead and his flesh has rotted away.
Disney's The Black Hole has a disturbing example. The creepy cloaked and mute robots on the Cygnus continue to water and care for the hydroponics bay (itself alive and almost overgrown) despite the entire crew abandoning ship leaving Dr. Reinhardt all alone. That they continue to care for it when all it does is feed one man and filter out his CO2 is seen as suspicious by the crew of the Pallomino. It's revealed that they former crew have been subject to Unwilling Roboticisation and the bay is used to feed them as well. The tragedy of this is brought to a head when the Cygnus is being pummeled by asteroids and none of the cloaked robots react to preserve their lives at all, and instead continue manning their posts. It's made poetically ironic when Dr. Reinhardt is trapped by a collapsing beam and begs for help, only to be ignored by the cloaked robots.
A famous human example is Charlie Chaplin in Modern Times where he goes berserk working on an assembly line tightening bolts in an ever accelerating conveyor belt. He eventually gets caught inside the machinery (where even there he's busy tightening bolts), and after he gets rescued he continues going through the motions, tweaking noses and buttons with wrenches on both hands.
In Sky Captain and the World of Tomorrow, Dr. Totenkopf's machines carry on his work of assembling a "Noah's Ark"-type rocket and loading animals on it despite him having died 20 years prior.
Used in Ciaphas Cain: Cain's Last Stand. After Warmaster Varanis killed by Cain, his shuttle pilot is found starved to death in his cockpit. Varan's main superpower is psionic brainwashing, and the investigators surmise that he ordered the pilot to wait for further orders and thanks to Cain, could never give him different ones.
Golems on the Discworld. If not attended, they will continue carrying out their last order indefinitely, potentially causing huge property damage. Other characters have mused that this is their approach to protest.
Played for laughs in Star Wars: The Essential Guide to Droids, which tells an anecdote where a binary load-lifter, a barely sentient droid that amounts to a forklift with legs, continued to stack boxes on a section of floor despite increasing signs that it was about to give way. After it collapsed onto the floor below, the load lifter just got back up and went to get more boxes.
In the Ray Bradbury short story "There Will Come Soft Rains" (known to many through the Fallout 3 location "McClellan family townhome"), a fully-automated house keeps performing its duties of cleaning the house, preparing meals, singing lullabies for the kids etc., even though the home has been empty for a long time and the family and everyone else has perished in a nuclear war.
In the Isaac Asimov short story "Risk" a robot pilot is to test a hyperspace drive and is given instructions to "pull the stick back firmly - firmly" until the drive engages. The drive doesn't engage, so the robot is stuck in that position and its human operators have to try to get it to stop but it just won't stop pulling because the drive hasn't engaged.
On the Babylon 5 episode "A Tragedy of Telepaths", Prime Minister Londo and G'Kar discover that G'Kar's former aide Na'Toth had been imprisoned and forgotten for the last two years in a Centauri dungeon, since nobody ever countermanded the late Cartagia's orders putting her there. Londo explained that that sort of thing happens with an absolute monarchy, and related a story of a guard detail that was continuously posted at a spot in the Centauri palace gardens, on orders from an emperor 200 years ago to guard a special flower there that had long since perished.
Somewhat parodied in Blackadder II when Edmund replaces his faithful manservant with another.
Edmund: Well, Bob, welcome on board. Sorry Baldrick, any reason why you are still here?
Baldrick: Euh .. I've got nowhere to go, my lord.
Edmund: O surely you will be allowed to starve to death in one of the royal parks.
Baldrick: I've been in your service since I was two and a half, my lord.
Edmund: Well that is the why I am so utterly sick of the sight of you.
Baldrick: Couldn't I just stay here and do the same job but for no wages?
Edmund: Well, you know where you will have to live.
Baldrick: In the gutter.
Edmund: Yes. And you'll have to work a bit harder too.
Baldrick: Of course, my lord.
Edmund: All right. Go and get Bob's stuff in and chuck your filthy muck out into the street.
Baldrick: God bless you, sweet master!
The cast of Red Dwarf first encounter Kryten obediently serving the three female crew members of the Nova 5, completely oblivious to the fact that they were killed when the Nova crash-landed.
In the Star Trek: The Next Generation episode "The Royale" an alien race had created a simulation of the Royale hotel from a potboiler book of the same name in order to bring comfort to a human astronaut which they had accidentally stranded on their world, having accidentally destroyed his ship and killed the rest of his crew. However, even after the astronaut died (hundreds of years ago) the simulation continues.
This was often the case in Gamma World. Any Ancient site that wasn't destroyed by the holocaust had robots continuing to perform the functions they did before the end. These included guarding the place, producing items and so on.
Many robots in the Fallout universe are unaware of the nuclear war that devastated America in 2077, and are still trying to carry out the tasks assigned to them in the Pre-War years.
In The Legend of Zelda: Skyward Sword, there are these cute Aztec-looking little electric robot guys in the Lanayru Mines. It's out of use, and the robots old stones by now, but Link reactivates the time stones the robots once harvested, which causes things to return to the way they once were (in certain spots). They are proud, single-minded laborers even after all those years.
In the firstMonkey Island game, Guybrush trains a monkey to hold down a switch so that he can enter a giant monkey head totem and descend into the underworld. Three games later, he returns to the area and finds the grave of that monkey, and is told that it waited patiently for him to return until it starved to death.
In Digger, the Dead God Underground has so-called "cold servants", implied to be vampires, which ceaselessly force its heart to beat in order to keep it alive.
The Friendship is Witchcraft episode "The Perfect Swarm" has a running gag with one particular pony watering a single flowerbed for hours on end. Even as a disaster is destroying Ponyville around her. Several episodes (and eight in-universe months) later, "Foaly Matripony" reveals that this pony is still watering those same flowers.
In the Looney Tunes short "Southern Fried Rabbit", Bugs Bunny encounters Yosemite Sam as a Confederate soldier guarding the Mason-Dixon line eighty odd years after the end of the American Civil War. When informed of this fact, Sam replies "I ain't no clockwatcher!"
WALL•E keeps trying to clean up Earth's surface after all humans have left even though he is the only robot still functioning and no real progress has been made in several hundred years.
In The Simpsons episode "Simpson and Delilah", Homer uses a hair grower to get his hair back and is promoted and gets an assistant who then takes the blame for something Homer did, gets fired, and still writes Homer's speech for him even after having been fired. The assistant is absurdly faithful to Homer.
Played straight with "The Sorcerer's Apprentice" short in Fantasia. Mickey, as the Apprentice, sets a magic broom to the task of fetching water from a well and pouring it into a cauldron, then goes to sleep and wakes to the room flooded with water since he never told the broom to stop. Then he finds he can't stop it and when he tries chopping the broom to bits, every bit becomes a new broom, all "programmed" to fetch water and throw it into the cauldron. It takes the return of the Sorcerer himself to stop the brooms (and save the apprentice from drowning).
Some fanatical Japanese soldiers continued to "fight" World War II on secluded islands in the Pacific after the war had come to an end, with at least one holding out for decades in the jungle, dismissing all calls for surrender from authorities as enemy propaganda.
Five hats means that five tropers think it is ready to publish.
You are saying that you think this draft is ready to be published. That means the description is not ambiguous,
it doesn't duplicate an existing trope, there are at least three examples, and the title makes sense.
Is that what you meant to do?
You are saying this draft has a ready-to-publish hat it does not deserve and you are taking it back.