"My lord," it was the master of the goblin blacksmiths, offering to build the king a new army, a mechanical army "seventy times seventy soldiers that can never tire, never know hunger or pain, and never be stopped."
The synthetic humans in Biomega come pretty darn close. They can live for months without food and for weeks without water and not lose efficiency.
Androids 17 and 18 in Dragon Ball Z are "infinite energy models" according to their creator, Dr. Gero. Androids 19 and 20 must absorb energy from people instead. This "Infinite energy" is also their weakness, since they can't exceed their power level unlike Cell or Gero.
Alucard, a vampire, in Hellsing was sealed in the Hellsing estate for decades without blood and was still "alive". Granted, greatly weakened, and dessicated into a husk, but he still woke up when exposed to blood.
Chiropterans in Blood+ won't die from starvation, as happened to one imprisoned in a concrete block in Russia. It was emaciated but still awake and capable of killing.
The eponymous warriors in Claymore do not need to eat or drink except in small doses. Though, when they awaken, they eat human flesh regularly, but can go without for a long time (several years), but it does weaken them...
The Angels in Neon Genesis Evangelion are explicitly stated to have an infinite power source in their bodies: the S2 Organ. Therefore, while alive, they have no need for sustenance, rest, etc. and are effectively immortal, though they can still be killed if damaged enough.
The Titans from Attack on Titan. While lack of sunlight and fatigue can slow them down, they do not age, sleep, or thirst and do not need the human flesh they eat note They lack digestive tracts beyond their stomachs and just regurgitate when full. They even regenerate biomass instantly and out of nowhere while thriving.
The Incredible Hulk cannot be stopped except by being calmed down by a close friend or loved one. Trying to subdue him by physical means only makes him angrier and his strength is proportional to his anger and has no upper limit.
Scrooge McDuck once had an immortal zombie (called Bombie) go after him for very long time, often walking nonstop across oceans, being frozen in ice for long time, and so on.
The zombies in The Walking Dead are Perpetual Motion Monsters, like most zombies. They technically don't last forever, as the characters have started to notice that more and more Walkers are extremely rotted, and on the verge of falling apart.
For X-Men foe Juggernaut, this is basically his power (mystical power in the comics, mutant power in the movie).
Taken to an interesting new height in Fear Itself. While possessed, he truly becomes unstoppable. At some point in the event, he calmly walks to San Francisco from a dozen miles away, while the X-men try a bajillion plans to stop him. Teleporting him away does nothing, he just reappears instantly. Digging a giant trench in front of him does nothing, somehow he strolls calmly across thin air!
Willy Pete from Empowered is an incredibly powerful fire elemental that eats superhuman flesh because he likes the taste, and because super-strong flesh is the only food that his mouth won't instantly incinerate. Being a fire elemental, he doesn't actually need to eat.
The eponymous army in Hellboy II: The Golden Army is a positively inhuman combination of perpetual motion monster, The Juggernaut, and regenerating. It was an ancient weapon Gone Horribly Right to the point the elf king who commissioned it sued for peace after it massacred a human army.
The monster from the movie version of Peter Benchley's Creature survived locked in a cage on the sea floor without any food or room to move for decades without any debilitating effects. The book version averts this by having it stalk out of its cage in a deep sea trench ever so occasionally.
The original Terminator movie made the eponymous cyborg sound like this trope, until the second film subverted it by revealing that his power cell would eventually run out....after 120 years.
In the Underworld film series, the first Lycan William Corvus was sealed in a tomb for 600 years. Once he's out of his coffin, he's at full strength. He clearly wasn't in suspended animation.
In The Lord of the Rings there's the Watcher in the Water, a massive squid-like beast living in the pool of water just outside Moria. It had been driven from the deeps of Moria into the flooded valley as a guardian and to trap the dwarves inside Moria. A creature that big just hanging around in caves needs to have SOME of this going on though.
In The Zombie Survival Guide zombies are explained by the virus which creates them replacing their mitochondria and taking over respiratory functions so they no longer need to eat or breathe (...somehow) and slightly more plausibly, delaying decomposition by protecting the body from rotting as the immune system would have. The later part of the trope is also averted, since zombies do rot away after a few years, unless they're in arid environments or end up being frozen.
Not "monsters" per se, but DiscworldGolems just keep going and going and going and going. They're used for menial, tedious labor typically, since they don't have to sleep and, for the most part, enjoy it as much as they can enjoy anything. They even use this as a form of passive-aggressive protest via Literal Genie; they're smart enough to take initiative and can read the intent behind orders, but if you mistreat them, they're liable to fight back by doing exactly what you say, constantly, until someone figures it out and tells them to stop. This can mean that the golem you asked to "pump some water" doesn't stop until your shop's street is partly underwater.
A more traditional version, also a golem, is Mr. Pump in Going Postal. As Moist's parole officer, it's explained that even though he can only walk while Moist can run, get a horse, or a cart, he will never tire, and never stop until he catches up to him. After one demonstration of this, Moist gives up on trying to run away.
You can hide, but you can't run!
In The Synthetic Men of Mars, a malfunction at an already imperfectly working cloning facility produces a constantly growing mutant entity that consists of a huge blob of human flesh and random body parts. It's stated that if it hadn't beenfire-bombed from above until there was nothing left, it could have grown forever by eating itself. Just think of the potential applications of something like that on a starving planet.
In Mortal Engines, the cyborg zombie Stalkers are explicitly stated to draw energy from the environment somehow, and are otherwise capable of functioning without needing to eat or recharge or refuel for an indefinite amount of time. And the oldest Stalkers, the ones made with Lost Technology, can't even die. They just go dormant, ready to fight again as soon as they're repaired and reactivated.
Lampshaded at one point; when a stalker expert is asked by Fever Crumb(who is rather confused about how Stalkers are possible) how they work, she responds with "molecular clockwork". Fever instantly identifies it as an In-Universe Hand Wave.
"That is not dead which can eternal lie..."Lots of examples in the works of H.P. Lovecraft.
In Firebird (Lackey), the dragon and the "dolls" in the Katschei's palace are implied to be mechanical in nature, possibly even robots that obey short logical orders. More than that, neither need sleep or fuel, though the dolls do seem to "tire" and need a day to recharge.
In the Left Behind series, one of the perks Nicolae Carpathia gets from being possessed by Satan is that he no longer needs to sleep, drink or eat; the evil power of Satan providing all of his needs. When Satan leaves his body, all of this is immediately canceled and Nicolae becomes a shriveled, emaciated barely-living cadaver.
The Jem Hadar in Star Trek: Deep Space Nine are genetically engineered so that the only source of nourishment they need is Ketracel White, a narcotic which keeps them loyal and without which they die after going violently insane. On top of that, they are conditioned to be unwaveringly loyal (though this has proven to be slightly unreliable, hence the White as a secondary measure).
The serpent that guarded the Golden Fleece, which never slept or left the tree except when drugged.
While similar, living constructs, the Warforged in particular are this as well, though it was changed to needing down time similar to sleep, though less of it than other races.
Artifacts in Magic: The Gathering usually have this quality, as do a lot of undead. Phyrexians in particular are prone to combining the two.
Role Master has the Black Reaper, an extremely powerful demon that is commonly used to guard items or places. Once disturbed, it will never stop pursuing the offender until he or she is dead. It is immortal and doesn't need food or air, and is never bored. When it follows it's soon-to-be victim, it will just walk underwater through oceans or use it's magic axe to tunnel through all but the hardest material.
Vampires in Vampire: The Requiem need blood to awaken in the evening but not to function normally. In theory, one could be perfectly happy in a sun-proof room for as long as it can force itself to stay awake at dawn.
One particular covenant of vampires, the Ordo Dracul, have a way to increase the amount of sustenance from the blood they drink, increasing to 1.5X what they get when they drink from humans and doubling blood drunk from vampires. Two such vampires can (canonically!) make infinite blood by feeding off of each other, becoming this trope. The downside is it leads to blood addiction and/or More than Mind Control and/or Food Chain of Evil kicking in. So to keep its members from trying this the elders in the covenant seal all such vampires in steel boxes which they dump into the ocean, whereupon said vampires spend eternity feeding from each other note (unless one eats the other, at which point it's an eternity of starvation).
Necromorphs in Dead Space don't need to eat, drink, sleep, or breathe. Have fun! The Hunter has the added benefit of infinite biomass to regrow lost limbs, making it almost unstoppable.
Dwarf Fortress actually has tags for these including [NOEXERT], [NO_EAT], [NO_DRINK], and [NO_SLEEP]. In-organic creatures like bronze colossi and iron men don't need any, while Goblins just don't need to eat or drink (due to the idea that all of them should die a violent death).
Alex Mercer and James Heller in the Prototype games are stated to have no need for food or rest and only absorb people to heal major wounds.
Reapers in Mass Effect can operate a seemingly infinite amount of time without the resupplying other ships need constantly. This gives them a rather considerable advantage against everyone else.
Left 4 Dead hangs a dramatic lampshade regarding the Infected. One of the pieces of graffiti on a safe room reads:
They don't eat, they don't sleep, WHAT KEEPS THEM GOING?
The Uncroaked/decrypted in Erfworld have zero upkeep.
Interestingly, the latter still eat and drink, and are provided with what they need as if they had their usual upkeep cost (in the textual interludes, Parson tried Decrypted archon rations). Something outside the known forces of the world is evidently picking up the tab.
The God Tiers in Homestuck are likely this. A player who reaches god tier can only die a heroic or just death. Death by starvation/thirst probably doesn't count.
They could go on a hunger strike, or be executed by starvation.
The wooden soldiers from Doraleous and Associates. Though they can be disabled with enough damage, their master need only command them to rise again and they will.