Whatever my one vulnerability is, I will fake a different one. For example, ordering all mirrors removed from the palace, screaming and flinching whenever someone accidentally holds up a mirror, etc. In the climax when the hero whips out a mirror and thrusts it at my face, my reaction will be 'Hmm... I think I need a shave.'
In a flashback in the first episode of Moshidora, Minami, playing in a junior baseball game, intentionally makes a terrible wild swing at the first pitch she faces, letting the pitcher think she is a terrible batter who will be easy to get out. On his next pitch, she makes a solid hit.
In Hero Man, the Big Bad implants some information about his race in the mind of a guy who later does a Heel Face Turn. This information includes the fact that the massive eye on his chest is a weak point that can be targeted to kill him. The titular robot punches said eye, putting his fist right through it — at which point the eye socket clamps down on his arm, immobilizing him long enough for the Big Bad to outright kill him.
There was one X-Men story where people discover ancient scrolls with rituals that would destroy Apocalypse. It turns out that they were created by Apocalypse himself, just to get people to try them out.
Subversion in Blackest Night, the heroes were led to believe that they had to recreate the white light of creation to stop Nekron, but their first attempt only made him stronger leading the heroes to believe they'd fallen victim to this trope. Turns out they needed to free the white entity itself to do the job right (and resurrect Nekron's undead anchor, Black Hand.)
Film - Animated
In the film Megamind, Metroman pretends to be vulnerable to copper to fake his own death, because he has come to feel he and Megamind are in a rut and he wants to explore another career.
This comes to bite Megamind in the ass later. It turns out, a trap made of solid copper can't stop the rampaging superhero-turned-villain Tighten.
In the film Seven Samurai the lead samurai says, "A good fort needs a gap. The enemy must be lured in. So we can attack them. If we only defend, we lose the war."
In The Bible, Samson made up quite a few of these to mask his true Weaksauce Weakness of cutting his hair. However, it was less an attempt to be Genre Savvy and more an attempt to get Delilah to stop pestering him about it. Upon learning each "weakness," Delilah would send some Philistine soldiers to try it out, only for Samson to laugh and kill them. He apparently learned absolutely nothing from this, and eventually told Delilah his real weakness.
In Carpe Jugulum the new (for a vampire) Count de Magpyre contributed to several holy books, giving them fake banishment rituals. This attempt may have been effective once, but hundreds of years later hardly anyone used them, possibly because they never actually worked, so he also worked very hard to build up a resistance to actual vampiric weaknesses like sunlight, holy water and symbols, and anal retentiveness.
This backfires on him and his family when they get a sudden vulnerability to religious symbols, as they appear everywhere in a world of thousands of religions.
One Animorphs book featured aliens with translucent skin and completely visible internal organs. When Jake has to fight them he realizes that no animal would evolve such perfect targets for a predator and deduces that they must be distractions. He hits one of the aliens in the one empty spot and it drops almost instantly..
It is implied in The Madness Season that many of the weaknesses that popular culture says belong to vampires were fabricated by their kind so that they could escape angry mobs relatively unharmed. Of particular interest is their "weakness" to wooden stakes. A properly skilled immortal is able to absorb the organic matter in the wood before it causes any damage to their heart. Unfortunately, some vampires wind up assuming that these weaknesses are the real deal, giving them a genuine aversion to such mundane things as garlic and holy water.
Late in the novel cyberpunk classic Neuromancer, Peter Riviera tries to get an edge in a battle with Hideo, 3Jane's guardian ninja by showing his illusion generators have a second function: they can concentrate their light energy into attack lasers…which he then uses to blow out said ninja's eyes. Hideo reacts for a second…but then calms down again. What Peter didn't know was that Hideo also trained himself as a Blind Weaponmaster.
Live Action TV
On The Vampire Diaries, Damon stabs Mason Lockwood with a silver knife, which backfires when it turns out that werewolves aren’t actually vulnerable to silver, and he just made an enemy of someone who wanted to live in peace. And eventually we find its more than fake—silver jumpstarts their healing factor.
An episode of Scrubs has the Janitor convincing JD that he goes into a trance when he hears the word "popsicle". Of course this just serves as another way to lull JD into a false sense of security.
Sylar does this in the third season of Heroes. Danko stabs him in the back of the head when he betrays him, which is the only spot that can stop the regenerating man via putting him in a coma as long as something is stabbed back there. Sylar gets up, though, thanks to his shape-shifting powers he got with Danko's help. He can now move his weak spot anywhere on his body, as he gloats.
Perhaps not the straightest example, but about halfway through Paper Mario, Bowser flat out asks Peach about Mario's weaknesses. If Peach answers honestly, there will be several powerful enemies around the next game area, but if she tells Bowser that Mario is deathly afraid of healing items, he will take this at face value, and there will be several useful items scattered around the area instead.
One enemy in Final Fantasy IV (found in the last dungeon) casts an Enemy Scan on itself each turn. This is all it does. The scan shows you its Hit Points (a little over the 10000 Damage Cap) and that it's weak to lightning. Cast a lightning spell on it and while you do actually do more damage, it causes the creature to Turn Red and unleash a brutal counterattack. Though, anyone who trusts an enemy named "Trickster" deserves what they get.
In the Monster Hunter series, the Gypceros will sometimes stagger and fall to the ground as if dead, only to thrash about moments later, most likely causing damage to any player that had attempted to move in to carve the corpse up.
In later games, it actually is possible to carve the Gypceros when it plays dead. It just isn't a particularly smart thing to do because of the aforementioned thrash attack.
This trope is inverted by Archer from Fate/stay night in the "Unlimited Blade Works" route when he gets into a fight with a 'not-holding back' Lancer. Lancer attacks and blocks faster than Archer can react, so Archer spends the entire fight on the defensive and creates intentional weak points in his defense in order to predict where Lancer will attack next in advance and block the attacks ahead of time. These are not fake weaknesses, however — Archer comments to himself how, if he misses a single block, he'll end up Impaled with Extreme Prejudice. Without these weak points he'll die a certain Death By A Thousand Cuts instead. Archer's objective isn't to win anyway, it's merely to drag out the fight long enough for the heroes to defeat Caster.
Subverted in Jade Empire, where multiple characters throughout the first two thirds of the game will comment on how your character seems to have a flaw in his/her fighting style that they distracted themselves trying to exploit while you whupped their asses. All of them conclude that it is merely a clever ruse and compliment your master for such ingenuity in teaching you. Thing is, once your master reveals himself as the Big Bad, he proceeds to show you how to PROPERLY exploit the weakness in your character's style.
In Scott Pilgrim, both Natalie Adams and Todd Ingram have powerful counterattacks if the player tries to attack them while they're knocked down.
The final boss in Golden Axe also counterattacks the player if attacked while knocked down.
Comes up in Sharin No Kuni. Hozuki's limp AND Kenichi's drug addiction. They're both faking them to get the other's guard down, and it works in both cases.
Shao Kahn of Mortal Kombat does a taunt that makes him seem vulnerable. Attack him during this, however, and he'll nail you with a painful counterattack.
The player might be able to do that with certain characters in Street Fighter Alpha. Stand just out of range, taunt, then strike them when they move into range. Works best with someone with long range like Vega (claw).
The Bonus Boss Contrarian King in Persona 4 starts off the battle by casting Red Wall on himself, which makes the target strong against fire attacks: enemies with elemental weaknesses generally try to cover them by casting the appropriate Wall spell on themselves, and taking advantage of them is strongly encouraged by the game. However, once it wears off and you try to cast a fire spell on him, he just absorbs it, causing you to both heal him and waste your turn. Though, like Trickster above, if you take a boss with that name at face value, you deserve what you get.
ST Arachnus from Metal Combat: Falcon's Revenge will sometimes reveal a hidden weak spot in its abdomen after you hit it with a powerful attack. While attacking this spot does deal a considerable amount of damage, destroying it also opens you up to a barrage of bullets from the lower body that is almost impossible to defend against.
Chrono Trigger: it can't be certain if if did this on purpose, but in the final stage of the battle with Lavos the core you need to destroy is not the vaguely humanoid part in the center, that's just another part that will regenerate after being destroyed. The real core is the limb-like part on the left, which looks identical to the non-essential part on the right.
Done in Sakura Taisen V by Ranmaru. During your second fight with him, the head of his mech is no longer his weak point and acts only as a distraction. The demonic bunny learned from his previous attempt. This time, he's hiding between the main body and the propellers.
In Spacetrawler, Yuri gets captured by some not-too-bright alien mercenaries who intend to torture her but know nothing about human physiology. She pretends to be horrified at the prospect of eating chocolate or butterscotch, so the aliens duly try to torture her with these. In the following pages she is given more and more "tortures" of the same vein which is then finally subverted as they cut off her limbs, they remain convinced throughout that that chocolate & the like was actually torturous to her.
A small one appeared in The Salvation War. A group of human insurgents were wiping out small groups of demons and their outposts, but always retreated when the Grand Duke of the area came with his army, making him assume that the humans feared him and his presence would always win battles. The fact he was leading his force made him easy pickings in an ambush.