D-list villain Whirlwind verbally demoralized Hank Pym to the point that Hank put a gun to his head in a once-controversial West Coast Avengers storyline. Only another superhero interrupting Hank prevented it from being a successful attempt.
This is how the Great Lakes Avengers defeated Maelstrom in their miniseries: Mr. immortal convinces the villain that life isn't worth living, and even "goes first" to cap his argument, prompting said villain to blast their own head off.
In Alan Moore's Captain Britain series, the sentient computer Mastermind once used holographic illusions of the Captain's dead parents who almost successfully urged him to kill himself out of guilt at their deaths.
Ms. Marvel: It is revealed that the selfish, amoral psychologist and psychiatrist Dr. Karla Sofen (AKA Moonstone) convinced depressed patients to kill themselves while she watched. Despite this, there have been a number of attempts to redeem her. If the characters knew the character the way the reader does, they would stop trying to redeem her, and either kill her or give her a Fate Worse than Death.
Pearls Before Swine: In one set of episodes, Rat takes a job as a late night radio show host, which means that he gets to listen to people call in to say that they abducted by aliens and things like. One such person calls from his truck to say that aliens took his brain. Rat responds by saying that the aliens took his brain because he's a smart fellow, that they'll want the rest of his organs for military use and he must not let that happen. The caller panics and asks what should he do. Rat tells him to drive off a cliff. The caller does it, and lets out an "AAAaaahhh" as he falls. One staff member points out the F.C.C. frowns on killing listeners and Rat says "Rules rules rules." This one is Played for Laughs.
Back To The Future Prequel: Someone (who Doc later realizes was the villain) calls Doc on Thanksgiving to say, "Why don't you kill yourself? Nobody likes you anyway."
In Canadian Bacon, the US city of Niagara pays its cops extra for cleaning up suicides. This led to unfortunate consequences...
Happens in Lethal Weapon multiple times. Murtaugh and Riggs are responding to a suicidal man standing on the ledge of a building. Riggs goes up to the roof in an attempt to talk the man down. After talking a bit, Riggs manages to get very close to the man and slaps a handcuff on him, handcuffing them together. The man starts freaking out, but Riggs actually starts encouraging the man to jump; he insults the man, saying that he's a coward for backing down now, just because his death will kill Riggs as well. Eventually Riggs jumps and pulls them both down...onto a crash-pad the police had already set up.
Immediately afterwards, Murtaugh, furious with Riggs, drags Riggs into a nearby building and they being arguing. Murtaugh thinks Riggs is suicidal and is a danger to himself and others. Murtaugh tells Riggs to just kill himself already. Riggs actually pulls out his gun and points it at his head, screaming at Murtaugh that he'll do it. Murtaugh one-ups this and yells back, telling him to go ahead and do it, since it'd be doing him a favor. Ultimately, Murtaugh is horrified when Riggs almost goes through with it. He assumed he was bluffing.
It's a Wonderful Life: Potter's cruel, heartless remark to George – "You're worth more dead than alive" – virtually drives our hero to the brink of suicide.
In Discworld, Ankh-Morpork citizens spying a potential building jumper will start shouting advice on the best buildings to jump from. Played for Laughs, (like virtually everything else) in Ankh-Morpork.
Charles Dickens uses this to firmly establish Scrooge as a Jerk Ass at the beginning of A Christmas Carol. When told that many of the poor would rather die than go to the hellish workhouses, Scrooge replies, "If they would rather die, they had better do it, and decrease the surplus population."
Back Story of Dune. While Liet Kynes is teaching the Fremen about his dream of making Arrakis a garden, a decision is made to kill him because he's a security risk to the sietch. A Fremen fighter is sent to execute him. When he approaches, Liet tells him "Remove yourself", and the man deliberately falls on his own crysknife. The other Fremen see this as an omen and decide to do anything Liet says.
When Richard is doing the trail in Neverwhere, various bill boards suggest throwing himself in front of the train would be a good idea.
In Rebecca, the Creepy Housekeeper Mrs Danvers very seriously encourages the second Mrs de Winter to commit suicide. That was because she was passionately devoted to the first Mrs de Winter and felt the successor was taking her place. (Mrs Danvers was, as you may presume, a total psycho.) She is not impolite or emotional when she does it, which makes it all the more scary.
In the Doctor Who episode "Dalek", the Ninth Doctor delivers this line almost verbatim to the titular creature, and gets a great big Shut Up, Hannibal! for his efforts. An unusual case of the (anti-)hero delivering this line to the villain.
The Doctor was also pretty insistant Lady Cassandra die, not just due to being a sociopathic mass murderer that takes over people's bodies, but he believes she has so greatly outlived her deserved lifespan.
On The Story Of Tracy Beaker (a live action children's show based on the series of books by Jacqueline Wilson) a character brushed off the main character by telling her to "go and play in the traffic".
In the first season of Dexter, one of the title character's targets was a Serial Killer that used this as his modus operandi; he was a psychologist who would encourage his vulnerable, mentally ill patients to kill themselves by taking an overdose.
Women on Maury who are looking for their babies' daddies sometimes say this when the possible daddy is particularly belligerent about paternity. Subverted in that the daddies (probably) aren't suicidal.
Benny: Think of next Saturday, going to Elland Road and seeing our beloved Leeds United play?
Man: I hate Leeds United. I'm a Liverpool fan.
Benny: Well, flaming well jump then!
M*A*S*H: Col. Potter deals with a suicidal patient by giving him the Radish Cure: Potter puts the mask from the knockout gas on the boy and forces him to continue to breathe in the fumes even when the boy tries to struggle free. Potter then points out the dichotomy, which makes the patient no longer suicidal.
The Bloodhound Gang song "Lift Your Head Up High (And Blow Your Brains Out)"
The song "Backmask Warning!" by Mindless Self Indulgence has this as it's central theme, with the chorus encouraging the listener to go kill themselves, with the verses featuring all kinds of horrific imagery that presumably would also encourage suicide:
Hate and devour the young and the weaker ones, and dont forget the guns
"Hey lobster. How do I make money off the internet?"
"Oh, oh. Kill yourself."
In Yu-Gi-Oh! The Abridged Series, Kaiba stands on the edge of a building, threatening that if Yugi attacks him he'll be thrown off by the shockwaves, meaning that Yugi has to throw the duel, as he did in canon. Unlike in canon, Yami tells him to go ahead. Later, Yugi sees him again, after having lost the duel:
Yugi: Hey, why don't you threaten to kill yourself again? Only this time, actually do it!
Lady Anne: Arise, dissembler; thought I wish thy death, I will not be the executioner
Gloucester: Then bid me kill myself, and I will do it.
Lady Anne: I have already
Gloucester: Tush, that was in thy rage; speak it again, and, even with the word, that hand, which, for thy love, did kill thy love, shall, for thy love, kill a far truer love; to both their deaths thou shalt be accessary.
Played for Laughs in Something Positive. Davan tells a group of diehard Rocky Horror fans that by protesting his Shock Treatment stage adaptation, they're only helping to create more publicity for the production. When one of the protesters says it hurts to hear that, Davan offers him "a little something to ease the pain." It's a poster of the Rippy the Razor mascot, showing how to slit one's wrists properly.
The Simpsons: When Homer deliberates a little too long in a line to jump off a skyscraper, the guy behind him says "Less chat, more splat, pal" and pushes him off. Subverted both in that Homer wasn't hurt by the fall, and because the guy was planning to do the same thing himself immediately afterwards, so he wasn't being a hypocrite.
Another example: The Season 19 episode "Eternal Moonshine of the Simpson Mind," where Homer can't remember what he did last day and he wrongly assumes that he hit Marge. He goes to a bridge to kill himself and Patty and Selma encourage him to do it, but when he goes to avert it, Patty and Selma push him off.
Lois shrewdly hints to Meg doing this in an episode of Family Guy.
Roger briefly implied Steve had to kill himself in American Dad! after unknowingly pleasuring himself with a nude painting of Hayley.
One episode of South Park has Cartman tell the hall monitor to kill himself.
Another episode has Stan calling a shopping network when his grandfather was spending all his savings buying from them, and, in a nod to the Bill Hicks example below, telling the presenters to kill themselves for taking advantage of old people. In the end, it works.