False Confession

A major crime has been committed and a killer is on the loose. Luckily for the police, someone has come forward confessing to the crime. Unluckily for the police, the person making the confession didn't commit the crime. Sometimes the police will see through the confession and instantly know that the person is lying. Other times, unfortunately, they will buy into the guy's story, derailing their investigation (at least temporarily). In some cases, even after the person's confession is shown to be false, some of the detectives will still go with it.

The possible reasons for the false confession are many. Perhaps the character is confessing because he knows or thinks one of his loved ones is really the criminal and is covering. Perhaps the character is mentally ill and is willing to confess to just about anything the police tell him to confess to. Perhaps the character just wants the publicity. And maybe, just maybe, the character actually thinks he committed the crime.

If the character has a history of confessing to things he didn't do, expect the police to remember him from the last time he made a confession. This could possibly lead to a subversion of this trope when it is later revealed that the "serial confessor" actually did the crime this time.

Taking the Heat is a subtrope. Often the product of the Jack Bauer Interrogation Technique.

Examples of False Confession include:

Anime & Manga

Happens in Paranoia Agent.
Inverted in a murder case in an episode of Detective Conan. The arrest was a ploy to get the killer to retrieve the murder weapon he hid (That no one else knew about) after the case is supposedly solved and get caught red handed.

Comic Books

In Watchmen, Night Owl and Silk Specter discuss a "supervillain" who would compulsively confess to all sorts of crimes because he wanted to be beaten up by a superhero.

"He pulled it on Rorschach, and Rorschach dropped him down an elevator shaft."

From Hell, apart from the dozens of letters from "Jack" (with different handwritings), the appendix accounts a few false confessions.
The reason that Eddie Brock (Venom) hated Spider-Man was because after Eddie broke journalistic ethics to apprehend someone who had confessed to being the serial killer Sin Eater, Spider-Man caught the real Sin Eater, revealing Brock's catch to be a false confessor.
Sin City has two rather tragic examples. Both John Hartigan and Marv are framed for crimes they didn't commit but in both cases, they confess to the charges to protect loved ones.

Film

In the 1931 film M, several people confess to being the serial killer, mostly for the publicity.
In White Heat, Cody confesses to a minor a robbery in Illinois he didn't commit in order to provide an alibi for a murder he did commit in California.
In Citizen X, a mentally disabled man leads the police on a wild goose chase after they get him to "confess" that he was the killer. The police pretty much know the man isn't guilty, but the political leadership want a scapegoat, even after more bodies pile up.
In Turk 182 a whole crowd of people show up to confess that they are the titular graffiti artist, including Jimmy Lynch (who really is Turk 182). It is, in fact, Jimmy?s disgust and derision at the false confessors that convinces a TV reporter that he is, in fact, actually Turk.
From David Fincher's Zodiac:

"I'm the Zodiac Killer."
"Okay, so how did you kill your victims?"
"With a gun... no, wait -- a hammer!"

In Election, Tammy covers up for Tracy by lying and saying that she tore down Paul's campaign posters.
In So I Married an Axe Murderer a person gives a confession for the "Mrs. X" murders, giving Charles the peace of mind to go through with his wedding to who he suspected was "Mrs. X". It turns out it was a false confession given by a mentally ill person, who also confessed to murdering Abraham Lincoln.
The Exorcist 3 has a mention of when a serial killer was on the loose, and the police wanted to weed out false confessions. So they released a false detail of the crimes, and everyone who confessed gave that false detail.

Literature

"Done It" Duncan from Discworld, who will confess to anything and everything (up to and including stealing fire from the gods). Can actually be a good source of information: if he says "It wasn't Bob who did that burglary, it was me", you can be pretty much sure that is was Bob who did it.
- Another Discworld example: in Feet of Clay, a golem turns itself in and confesses to a murder. Colon is ecstatic when the golem also confesses to every unsolved crime in the city. Carrot demonstrates the problem with this by getting confessions to things that never happened.
Dorothy L. Sayers' Montague Egg story, "Murder at Pentecost"
In Purple Hibiscus Jaja gives one of these to protect his mother, who has just killed her abusive husband.
Subverted in Murder at the Vicarage, where the people who initially confessed to the crime were actually the ones who did it.
The publicity version is parodied in The Long Dark Tea-Time of the Soul, when a Heathrow check-in desk blows up:

The usual people tried to claim responsibility. First the IRA, then the PLO and the Gas Board. Even British Nuclear Fuels rushed out a statement to the effect that the situation was completely under control, that it was a one in a million chance, that there was hardly any radioactive leakage at all and that the site of the explosion would make a nice location for a day out with the kids and a picnic, before finally having to admit that it wasn't actually anything to do with them at all.

Live Action TV

On Dexter, Neil Perry confessed to being the Ice Truck Killer because he wanted to be someone important.
In The Unusuals, a character is established in the backstory as a serial confessor. It turns out that he would falsely confess to crimes so that he could get inside the police station, where he was involved in a scheme to steal evidence.
Law and Order had several of these.
Criminal Minds featured this at least once.
Barney Miller had a background character who compulsively confessed to all sorts of crimes. No one ever believed him.
Subverted on an episode of Monk, where a woman repeatedly comes into the police station to confess to "crimes" such as murdering her roommate's gerbil. It turns out she was searching each of the interrogation rooms in turn trying to find a diamond that had been hidden under one of the tables.
This is a major part in the Lily Kane murder mystery in the first season of Veronica Mars: Disgruntled former Kane Software employee Abel Koontz confesses to the murder of Lily Kane, but in reality the Kanes agreed to pay for his daughter's future so that he would take the fall (He's dying from a disease, so he doesn't particularly care what happens to him), because they believe that their son Duncan killed Lily in an epileptic fit. Because of his confession, Sheriff Lamb ends the investigation, saying they found the killer, causing Veronica and her disgraced-Sheriff-now-PI father to investigate it themselves.
A man in the CSI episode "The Descent of Man" confesses to the murder of his spiritual guru because one of their tenets is being extremely agreeable.
A CSI New York episode featured the case of a man walking into a police station confessing to have murdered a doctor, carrying the revolver used to kill her as proof and gunshot residue on his clothes. Turns out it was his wife that killed her. The man has a terminal illness and the couple was swindled by the doctor into spending all their money in a non- working treatment, causing the wife to kill her in a rage, leading the guy to discharge the gun someplace else to incriminate himself. The episode ends with the man pleading to Stella that she arrest him instead seeing as he is about to die anyway, only for her to tell him she can't.
An episode of The Thin Blue Line had a guy being arrested after confessing to the crime the team was investigating. When he is interrogated he confesses to the crime...along with several others...and starting WWII...
One episode of Columbo ended with a friend of the killer confessing. As the killer is dying of a brain tumor (and in fact, as a result of the tumor doesn't even remember that she's the killer), Columbo, who knows the truth and that the victim deserved it, gently points out that in the hands of a good defense attorney the evidence will show that the friend didn't do it. The man agrees, but says it will probably take a good six months of so before he'll go to trial, by which time the woman will be dead.
New Tricks had a man come in and confess to the killings of numerous dogs. The man has been the prime suspect in that crime for years and is also confessing to a string of new dog killings. However, by that time the detectives have determined that hew did not do it and easily trick him into confessing to parts of the crime that did not happen. He wants publicity for his anti-animal views.
Life On Mars had several of these, including the factory worker confessing to what was revealed to be an accidental death.

Video Games

Ron Delite in Phoenix Wright Ace Attorney: Trials and Tribulations, a meek and timid guy, confessed being the Gentleman Thief Mask*deMasque the day after his last heist, even though he didn't have the object stolen, and when Phoenix goes to visit his wife she tells him that Ron is deluded and thinks he is deMasque. Turns out Delite is Mask*deMasque, but didn't commit the crime in question. He confessed to have an alibi because at the time of the theft there was a murder in which he was implicated, but which he didn't commit either, and the one who did commit the murder had the same idea and had Phoenix find him as Mask*deMasque in court.

Western Animation

A variation is used in the My Little Pony Friendship Is Magic episode "Party of One": Pinkie Pie, suspecting her friends are hiding something from her, forces Spike to tell her exactly what she wants to hear: that they're avoiding her because they don't want to be friends anymore. In this case, it's obvious that isn't really the case, and a stressed-out Spike is just parroting back her own words to make her stop pressuring him.
Happens in South Park when Mr. Mackey is investigating which of the boys crapped in the urinal, and thinks it's over when Clyde confesses. Then Clyde's parents arrive and inform him that he had a colostomy.
In The Simpsons' "Who Shot Mr. Burns?" 2-parter, Smithers remembers stumbling drunkenly around town and shooting at someone the night of the crime. While in custody, he's exonerated by evidence that he was at home when Mr. Burns was shot. Smithers did shoot Jasper - but it was in his wooden leg.

Real Life

A man named Robert Hubert confessed to starting the Great Fire of London in 1666 by throwing a fire bomb through a bakery window. It was shown during his trial that he was out of the country until two days after the start of the fire, that he was never near the bakery in question in any case, the bakery had no windows, and that Hubert himself was a cripple who was physically incapable of throwing a bomb. Despite all this, Hubert was found guilty and hanged for the crime.
Over a hundred people confessed to the Lindbergh Baby kidnapping in 1932.
- And over five hundred confessed to committing the Black Dahlia murder in 1947.
- Especially in high profile cases like these, it's now fairly common practice for police not to release a key detail about the crime scene or victim, so they can easily tell that a confession is false if the confessor doesn't know that detail.
John Mark Karr confessed to killing JonBenet Ramsey. Not only did he not do it, he had a record for being a serial confessor.
It's not just the whackjobs confessing: it's well documented that police can get false confessions out of people they honestly believe to be suspects by wearing them down in interrogations and/or scaring them. Despite all the times this happens, some juries, police and prosecutors still claim that innocent people wouldn't do this and express disbelief at the idea.
- It has been shown that even if the jury knows a confession is false they are more likely to convict a person who gave a false confession, than one who did not, with the same evidence.
Lee Strobel, during his job as a reporter for The Chicago Tribune once recalled an unusual case where a man pleaded guilty to shooting the host of a party. It was later found that the host had shot himself by accident, and the man "confessing" had done so because he found that his time being spent in jail while awaiting his trial counted as part of his sentence. If he was found guilty, he would get to go home anyway in three days. If he pleaded not guilty, but was found guilty, then his sentence could have been massively increased.
Sadly common in totalitarian regimes. An old Soviet joke runs like this: The CIA, the KGB and the GIGN have an argument about who is best at catching fugitives. Tired of their bickering, the Secretary-General of the United Nations sets them a task - each is given a ten square miles of forest, in which there is a white rabbit, which they must catch. Their respective efforts go like this:

CIA: The CIA bugged the entire forest, used satellite imagery, set up a network of informants among the woodland creatures, and interviewed all plant and mineral witnesses. After six months with no leads, they concluded that the rabbit does not exist.
GIGN: The GIGN, after a month with no leads, burned down the forest, killing everything in it. They argue that the rabbit had it coming.
KGB: Four KGB agents walk into the forest. Ten minutes later, they emerged with a badly tortured squirrel, which is yelling "okay, I'm a rabbit, I'm a rabbit!"