From Beauty and the Beast, the Beast becomes this after Belle leaves. When the castle is under attack, he outright says "Let them come." and it's only when he sees that Belle has returned that he fights back against Gaston.
In Igor, Scamper is a rabbit that Igor gave intelligence and immortality—to his annoyance, since Scamper wants to die. He gets grievously harmed numerous times throughout the movie, all while snarking about it, but winds up appreciating life more by the end.
Tigger becomes this in The Tigger Movie when Rabbit points out it's not safe to be in the snow.
Rabbit: Are you crazy? It's not safe out here. Tigger: Exactakly! That's why you should all go home where it is safe. But I've got to wait here in my family tree for MY! REAL! FAMILY!
In Up, there's a kind of variation Word of God said Carl's motivation was originally supposed to be using the balloons so he could basically be with Ellie "in heaven." Of course, you couldn't actually have him kill himself and there wasn't much of a story you could get of him just floating around in the sky being sad, so the plot gradually changed to him trying to reach Paradise Falls (where he and Ellie always dreamed of visiting) while finding new meaning for his life.
Peter B. Parker in Spider-Man: Into the Spider-Verse has elements of being the death-seeker; thinking that he's ruined any hope of a future with his MJ, he's more than willing to stay behind to shut down the super-collider and send the other Spider-people home, even knowing that he won't survive long in this universe. Miles Morales manages to convince him to go home.
Forrest Gump has Lt. Dan, who has ancestors who have fought, and died, in every major American war. He winds up making it out of The Vietnam War alive and becomes one of these, feeling like he's a failure for not continuing the "family tradition", so to speak. He gets over it after surviving Hurricane Carmen.
In the book's sequel Gump and Co., he gets killed by friendly fire during Operation Desert Storm.
Nick in The Deer Hunter is a Shell-Shocked Veteran traumatized by the terrors of the Vietnam war. He never recovered from an incident where he was forced to play a game of Russian Roulette for the amusement of his Viet Cong capturers. He ends up as an Empty Shell living out his life doing Russian Roulette for money in Saigon.
James Bond movie Skyfall. Oh yes, Bond killed the villain, but it's made fairly clear that Raoul Silva would've killed himself after completing his revenge on M.
Silva: So, I had only one thing left. My cyanide capsule in my back left molar. You remember, right? So, I broke the tooth and bit into the capsule. It...burned all my insides, but I didn't die. Life clung to me like a disease. And then I understood why I had survived. I needed to look in your eyes one last time.
Sgt. McCron in The Thin Red Line after his entire squad dies. (In the novel, he merely has a nervous breakdown.)
Baron: And that was only one of the many occasions on which I met my death, an experience which I don't hesitate strongly to recommend!
In the various The Prophecy movies, fallen angel Gabriel's unwillingly-revived henchpeople fall into this category: when one them gets (re-)fatally shot by a protagonist, his last comment is a sincere "Thanks pal, you're a sport!"
Doc Holliday in Tombstone. "Whelp...got the consumption...might as well shoot at people."
John Ryder from The Hitcher. As the plot uncovers, he repeatedly asks Jim Halsey to kill him in cold blood (after their first encounter when Jim picked up Ryder as an unsuspecting hitchhiker). When Jim fails to do so, John proceeds to go on a path of carnage.
Stargate: Following his son accidentally shooting himself with Jack's gun, Jack O'Neill undertakes what he figures is probably a suicide mission.
Dr. Daniel Jackson: [to Colonel O'Neil] I don't want to die. And your men don't want to die, and these people certainly don't want to die. It's a shame you're in such a hurry to.
Riggs in Lethal Weapon films start out this way, but his partnership with Murtaugh changes him for the better.
Louis, in the film version of Interview with the Vampire, becomes this when his young wife dies in childbirth and the baby doesn't survive either. His willingness to die leaves him open to Lestat's machinations. Note that this is a complete change from the original novel, in which Louis was unmarried.
Miles Tuck in Tuck Everlasting, due to losing his wife and children. He's fought in every war he can find and regularly picks fights, trying (unsuccessfully) to die.
Pirates of the Caribbean: Dead Man's Chest: Of the commanding officer's Survivor Guilt/Shell-Shocked Veteran variety, we have James Norrington in the opening. He'd deliberately steered his ship into a hurricane while chasing Jack Sparrow, resulting in the loss of the ship and most of the crew, and having to resign his commission. The next time we see him, he's a grimy drunk who seemed to be pretty zealous about his bar fights.
The WWI pilot in The Mummy (1999) wants to go out in a blaze of glory like the rest of his deceased war buddies. Seeing as he dies fighting a giant face made of sand while successfully escorting Rick and Johnathan to Hamunaptra, it's probably safe to say that he succeeded. His last words?
"Here I come, laddies!" [huge laugh]
Hell, it's how they sell him on taking the mission in the first place.
Winston: So, what's your little problem got to do with His Majesty's Royal Air Corps? Rick: Not a damn thing. Winston: Is it dangerous? Rick: Well, you probably won't live through it. Winston: By Jove! Do you really think so? Jonathan: Well, everybody else we've bumped into has died. Why not you?
Much like one of its main inspirations, The Dark Knight Rises puts Batman firmly in this territory. His world, his future, and his entire life has been shattered, so despite being in bad physical shape, he puts the cowl back on and throws himself into a fight with a much tougher opponent rather than find a way to live as Bruce Wayne. He finds his "fear of death" again and by the end of the film, he's able to give up being Batman and find a life for himself.
Sir Lancelot in Excalibur. He's more of a defeat seeker than a death seeker though, having traveled around looking for a King who was good enough to beat him and thereby win his fealty. He claims he was Cursed with Awesome.
J.B. Books in The Shootist. The death that was coming for him, though, was far worse than the death he sought.
In Man of Steel, Zod admits straight up he has nothing more to live for and goads Superman several times during their fight that he will have to kill him or be killed. Even his final act — trying to fry innocents with Heat Vision while in a choke hold — is basically pleading for Kal-El to end it.
Subtly hinted at with Riddick in Pitch Black. That trait is (mostly) ditched in later incarnations.
In The Grey Zone, most of the Sonderkommando can't live with what they've been forced to do — assisting the Nazis in disposing of the bodies from the gas chambers by cremating the bodies of other Jews, or they'll be killed as well. Simon discusses this quite extensively with Dr. Nyizli, stating that he doesn't want to live after everything is done. He chooses to die in the explosion which destroys the crematoria.
The Nature Of The Beast has this trait as a reveal in the final act. The main characters are a serial killer and a white-collar embezzler who know each other's secret. It seems like the meek wage-slave is the embezzler and the brash vagabond is the killer. However, it turns out that the vagabond is the embezzler, and he's been tormenting the serial killer wage-slave because he wants to be killed by him.
Captain America: The Winter Soldier ominously implies this about, of all people, Captain America. His 70-year sleep has robbed him of his entire world; he's a 20-something WWII vet stuck in an era where the girl he loves is 90 years old, long married, and is bedridden and suffering from Alzheimer's and later he finds he may have "died" for nothing — Hydra is alive and at the heart of SHIELD, and his childhood friend has been turned into a mindless weapon. When Sam Wilson asks him what makes him happy, Steve candidly tells him he doesn't know. Throughout the film, he takes one deadly risk after another and during the climax he gives Maria Hill the order to have the carriers fire on each other — despite him still being on one of them — and seems to resign himself to going down with the ship. He only springs back into action when he sees Bucky trapped under one of the girders, and when Bucky tries to kill him, he refuses to fight back.
Nathan Algren, due to the massacres of the Indians he had to carry out previously, which still haunt him. While training the fledgling modern Japanese army, he gets on the firing range and orders a recruit to shoot him (only to miss, predictably) to prove a point about the soldiers not being ready for combat yet. Later in Katsumoto's mountain village, he doesn't even react when a Samurai warrior threatens to decapitate him.
Katsumoto labels himself as one as well.
The recklessand rage-filled way Darth Vader/Anakin dueled with Obi-Wan Kenobi at the end of Revenge of the Sith could be explained by a combination of this trope and being drunk on the dark side. There were several times during that duel that Obi-Wan could have taken advantage of Anakin's complete lack of self-preservation before he actually did.
The War Boys in Mad Max: Fury Road believe that an impressive death is the way to Valhalla - such as hurling themselves onto an enemy vehicle holding a grenade-fitted spear while high as a kite on chrome spray paint and bellowing "WITNESS ME!" at the top of their lungs. Given that most of them have cancer or some other unpleasant side-effects of the post-apocalyptic setting - Nux makes reference to "night fevers", has visible tumours on his neck, and is in such bad shape early in the film that he needs to have Max's Type O blood piped in from the front of his car during the initial pursuit of Furiosa's war rig - it would probably be a comparatively noble way of sparing themselves a slow, painful death, if they weren't serving a weapons-grade asshole like Immortan Joe. Nux, unlike the others, gets his death in the form of a Heroic Sacrifice after his HeelFace Turn.
In The Last Witch Hunter, Kaulder, especially in the flashbacks, seeks death to reunite with his deceased wife and child. Unfortunately for him, he's immortal, but in the present day, he seems to have come to terms with this.
Murder in the First: Young asks Stamphill to change his plea to guilty and get him the death penalty several times, as he's afraid of going back into solitary otherwise, which he views (with reason) as a Fate Worse than Death.
Blazing Saddles features a variation. After nearly gunning down a kid, Jim gave up gun-fighting and tried to drink himself to death.
Bart: A man drink like that and he don't eat, he is going to die. Jim: (wistfully) When?
Gettysburg doesn't go into the details of the matter (see the Literature entry for The Killer Angels for that) but still clearly depicts Confederate general Dick Garnett as determined to die on the field of battle. General Armistead, aware of this, tries to get his superiors to intervene, but they can't decently do so because the Confederate officership is so obsessed with personal honor. Garnett nearly reaches "The Angle" and then appears to deliberately ride down a Union cannon; the next shot is of his horse, riderless.note The details are that Garnett had been accused of cowardice by his previous commander, Stonewall Jackson. Garnett then insisted on riding a horse when partaking in Pickett's Charge, which made him a sitting duck.
Falcon Rising: At the start of the film, John is suicidal, both considering using a gun on himself and interrupting a liquor store robbery by taunting one of the robbers to shoot him.
At one point, Wyatt puts the barrel of nailgun in his mouth as if on a whim and pauses for a long time before taking it out. He tries to put a nail through his hand, but the nailgun jams.
Christian admits to contemplating suicide in the previous year before seeing a therapist for a single session. In the climax, he volunteers to let Wyatt tie him up and gag him, knowing that Wyatt might murder him in a fit of paranoid psychosis.
Fearless (1993): After narrowly surviving the plane crash, Max believes that he's not truly alive. He keeps gambling with his life to test out his hypothesis, such as walking into traffic, dangling off the edge of a rooftop, and eating food that he's deathly allergic to. He only recovers at the end after going through a second Near-Death Experience.
Solomon Lane, the Big Bad of Mission: Impossible Rogue Nation, becomes this in Mission: Impossible Fallout after having been tortured and interrogated for two years. During the chase in Paris, he doesn't flinch when Ilsa tries to shoot him and in the climax, he stays behind rather than escape on a helicopter with Walker, planning to die when the nuclear bombs detonate.
At the start of Avengers: Endgame, Clint Barton loses his entire family when Thanos "snaps" half the universe out of existence. The five-year Time Skip shows Clint has now become the masked Ronin, dishing out brutal attacks on surviving crime cartels. It's obvious that Clint is risking his life so much on the hopes he dies and can be with his family. In fact, when Scott Lang warns of the dangers of a time travel experiment, Clint volunteers without hesitation, showing he doesn't mind if he doesn't come back. He even tries to sacrifice himself for the Soul Stone, arguing that after all he's done, he doesn't deserve to be happy with his family again.
Violet & Daisy: The Guy it turns out actually wants to die, as he's dying of terminal cancer. Because of that he stole from a criminal gang, knowing they'd have him killed for it.