Since the missing person who committed a crime is likely going to keep up with his story, have news stations/media ever given false information to the public in hopes of fooling the perpetrator to somehow make a mistake or turn himself in? Any famous example of this?
EDIT: Is this the right subreddit for this question?
Yes, this is frequently done. One of the most famous cases is that of Susan Smith. She was a young mother of two boys, recently split from her husband, who in 1994 claimed that a black man carjacked her and drove off with her two kids still in the car. The police suspected her story from the beginning, believing she'd already killed them. However, instead of revealing their suspicions, they just gave her enough rope to hang herself with, letting her cry and plead for the return of her kids on TV, all the while collecting evidence against her. It was top news every day for the whole week, with the general public falling completely for her tragic story. After 5 days or so, the truth came out: she'd driven her car into a lake with the boys inside. Her motive was likely that she was finding that having kids was an obstacle to dating. She's in prison for life: https://en.wikipedia.org/wiki/Susan_Smith. I'm not positive, but I think this story did more than any other case, to fix in the public's mind that in cases of mysterious child abduction, it was probably a parent that did it. You see the result just a few years later, when JonBenet Ramsey went missing. There was widespread belief that one or both parents did it. There still is such a belief, though the DNA of someone else was found on her.
Slightly different from your question, but yes, the FBI recently admitted to using a fake news article to track down a criminal.
The AP is pretty pissed off about it. Source
Also, it is common to hold back certain details from the media/public in order to be able to weed out fake confessions, since only the killer would know certain details.