In Monster, it turns out Johan did this to his sister Nina during the first episode, killing their foster parents and enticing Nina to shoot him. Obviously, she did. It eventually turns out he wants the protagonist Dr. Tenma (since he saved Johan and allowed him to continue being... who he is) to shoot and kill him to prove that human lives aren't equal.
Noir had Altena doing this. She seemed to think the titular duo killing her would result in them becoming The Scourge of God. It's not clear if what happened "counts."
Aion does this to a weakened Chrono after the latter is hit with a blast from Rosette's gun. Aion believes Chrono has grown soft.
In Trigun, Legato Bluesummers tries to goad Vash the Stampede into killing him. Vash is a Technical Pacifist who believes "no one has the right to take the life of another"; Legato knows that forcing Vash to betray his beliefs and kill him is the single most abhorrent thing he can do to Vash.
Legato: It's alright, kill me. It's simple. All you need do is pull the trigger. Once you've killed me, this will all be over. Come on. Time to choose. You have free will.
Naruto inverts this: Naruto says that if he can't get Sasuke to make a Heel Face Turn, then he's Taking You with Me. He's gotten Sasuke to promise to save all of his hatred for him and avoid killing anyone else until they fight.
Explicit in Fullmetal Alchemist's final showdown between Envy and Mustang. While not part of Envy's plan at all, the normally icy cool Mustang's allies are so horrified at the sadistic glee with which he's torturing Envy to death that they beg him to stop until he can realize what he's doing and cool off. This ultimately acts as something of a wakeup call that saves the character from the nihilistic death spiral he'd been in to that point.
Uttered by Yomi to Kagura at the end of Ga-Rei -Zero-. By that point, everything that Yomi cared about was destroyed, some of them by her own hands while being Brainwashed and Crazy, so she believed that death by the hands of her own adopted little sister is the only way she can retain some dignity out of the whole mess.
In the other, Naraku has protagonists split up within the final dungeon. He shows the warrior Sango that her love interest, the monk Miroku, is about to sacrifice himself to try to destroy Naraku, without realizing that it is only an illusion of Naraku that he would destroy. He uses the Anti-Hero Sesshoumaru's Morality Pet Rin as a Human Shield, and tries to get Sango to choose to kill Rin in order to destroy him and save Miroku, which would further corrupt the sacred Shikon Jewel with darkness (as well as probably turn Sesshoumaru against Sango and possibly the rest of the protagonists). To top it all off, unbeknownst to her, the Naraku speaking to Sango is actually just another illusion, so Sango wouldn't actually hurt him or save Miroku at all. The plan successfully tricks her into trying to sacrifice Rin, contributing to the corruption of the jewel, although the other protagonists manage to save Rin, and Sesshoumaru remains focused on Naraku as the enemy.
The Joker does this to Batman a lot. There was also one story where he did it to Superman - not just goading him, but making Clark believe that killing him was the only way to save Lois.
Joker:"Tonight you're gonna break your "one rule"."
Especially since The Dark Knight is partially based onThe Killing Joke, which involves the Joker trying to drive Commisioner Gordon to do this. He fails in both stories.
There was a Superman story arc where Manchester Black, a "superhero" from waaaaay down the cynical end of the scale messes with Superman's life in an attempt to get him to admit that idealism has its limits. Culminates with him (apparently) killing Lois Lane right in front of Superman, willing to accept the consequences because if Superman snaps and kills him that means he was right all along. He fails, of course.
Possibly played with in 8mm. Eddie Poole taunts Tom Welles, who has him at gunpoint, telling him he doesn't have the guts to pull the trigger - and he's right. Until Tom whips out his cell phone, calls the mother of the girl Eddie and his fellows killed for their Snuff Film, and lets her talk him into doing the deed.
In Changes Martin arranges that Susan learn he was the one who revealed her daughter's existence to the Red Court, so that she'll kill him and complete her transformation into a Red Court vampire. Very unusual in that Martin is a Well-Intentioned Extremist making a Heroic Sacrifice by doing so. This happens in a room where a bloodline curse has been prepared - when a person is killed as the focus of it the curse will destroy everyone who came before them. Since Susan's transformation happened seconds before she's the youngest of the Red Court, meaning that if she's killed this way the entire Court will be destroyed. He's trusting Harry to finish the job, and he does.
A thug called Roddy McGristle does this to Drizzt Do'Urden in Sojourn, but only after finding that a) he can't beat Drizzt and b) Drizzt can't kill him, leading to an impasse. Then Bruenor confronts him and he sees that he'd have no qualms about killing him. Or eating his dog's leg, apparently.
Legacy of the Dragokin: Jihadain does everything she can to provoke Daniar into breaking her rule against killing. The reason behind this is that only lethal rage from a pure soul can revive Kthonia.
In Smallville, Legion, Brainiac taunts Clark to kill him. Clark couldn't bring himself to do it because he is in Chloe's body.
In an early episode of Supernatural, Dean does this to a very pissed-off Sam who's had his head messed with by a ghost. Dean even hands Sam a gun and tells him to do it. Sam does. The gun isn't loaded. Dean's not stupid. In season 4, Lilith pulls this with Sam when he'd rather reunite with his brother than kill her like she wants him to so he'll inadvertently start the apocalypse.
A wily serial killer who Gibbs caught years before NCIS started asked to see Gibbs again days before his execution date, taunting and teasing that he'd tell where the bodies are kept. This is the guy who turned Gibbs from a jovial jokester like DiNozzo into the jaded guy he is in the show. He wants Gibbs to kill him in attempt to ruin Gibbs, since he's not even a day away from execution.
A rare heroic variant occurs in the Doctor Who episode "Victory of the Daleks," when the Doctor is trying to get the "Ironsides" that Winston Churchill insists are robots created by the Allied special forces to admit they are actually Daleks, only to be constantly told by them that they do not understand, that they are his soldier, and would he care for some tea? He finally flies into a violent rage and beats one of them with a giant wrench while screaming at them to show everyone how evil they really are. It's unclear if he's exhibiting Death Seeker behavior, if he's too angry to think about what he's doing, or too angry to care either way:
The Doctor: Fight back! I know you will! You hate me! You want to kill me! Well go on! Kill me! KILL ME!!
In Cold Blood, a Silurian called Aleya attempts to provoke her human captors into killing her, so that it will spark an interspecies war.
In Lost Girl, Lachlan's treatment of Lauren is basically one Kick the Dog moment after the other. Bo finally confronts him on it and seems hell bent on killing him. He eventually drops his sword and encourages her to do so. Naturally, she doesn't. He then reveals that his treatment of Lauren and goading her to kill him was a test. There is something much worse coming (that feeds off of anger), and he had to make sure she was up to the task of confronting it. He notably becomes more likeable afterward, and treats Lauren better.
In Kingdom Hearts Birth By Sleep, this is what Master Xehanort has been doing to Terra all the way up to their final confrontation in order to make sure Terra's heart is at it's weakest so that he can Body Surf into a younger, healthier body.
After being defeated by Ragna, a delighted Terumi laughs and and again calls out to Ragna to "strike him down with all of his hatred". Ragna has a tough decision, and finally refuses, deciding that there are plenty of things that can make killing Terumi a bitch.
At the end of the third Disc of Final Fantasy IX Kuja goads the party into attacking him specifically because he needs to harness their agression and put himself in a state of physical desperation so he can Trance with the accumulated energy of all the souls stolen from the Invincible, gaining god-like power
In Star Wars: The Force Unleashed, the Light Side Ending has Starkiller defeat Palpatine in a boss battle. Palpatine groans and says, "You were destined to destroy me. Do it. Give in to your hatred!" When Kota talks Starkiller down, Palpatine gets really pissed and blasts Kota with Force Lightning for interfering.
In the second game, Darth Vader dares Starkiller to kill him when Vader is defeated. If you try it, Vader's Dark Apprentice will jump in and kill Starkiller.
In Castlevania: Dawn of Sorrow, Celia attempts this on Soma Cruz, in a gambit to make him the Dark Lord. Whether she succeeds or not depends on if Soma has a certain amulet equipped. If she does, it leads to a Bad End and a new game play mode where Soma is the final boss.
In The Legend of Spyro: The Eternal Night, Gaul tells Spyro to finish him after being defeated. When Spyro doesn't attack, Gaul calls him a coward and laughs at his "weakness". Unfortunately for Gaul, while Spyro might not have finished him off, Dark Spyro is more than willing to and completely obliterates him.
In BlazBlue: Continuum Shift, Ragna has turned the tables on Hazama, beating his Blaz Blue after absorbing Nu. To Ragna's surprise, Hazama starts to egg him on. Deciding to take care of him later, Ragna knocks him out and goes to deal with Noel. It turns out that Hazama wanted Ragna to kill him, so he could go to his ghost form and invade Takamagahara. Instead, Relius shows up and finishes off Hazama himself.
In World of Warcraft: Mists of Pandaria, when you attack the Sha of Anger, a literal physical manifestation of anger, yells for all of the zone to hear "Yes, YES! Bring your rage to bear! Try to strike me down!"
Star Wars The Old Republic: Expected from the franchise at this point, but often well executed. Sith NPC's will regularly try to bait light side characters and especially jedi with this.
In Guiile's original ending from Street Fighter II, once Guile grabs Bison by the collar and yells at him about his murder of Charlie, Bison replies with this and urges Guile to kill him. It's a good thing that Guile's wife and daughter show up and tell him "If Youkill Him You Will Be Just Like Him" instead.
In The Order of the Stick, Belkar (a Chaotic Evil protagonist) tries to get Miko (a nominally Lawful Good antagonist) to do this, just so she'd lose her paladin powers. (And then Belkar could get resurrected). Vaarsuvius later points out to Belkar that the Order of the Stick lacks the resources to resurrect Belkar, anyway.