Possibly the earliest television example... in an episode of Dennis the Menace called "Dennis and the TV Set", Fix-it Man Opie continually does this.
In Psych, Shawn, who's been shot and is locked in a trunk, accidentally calls a woman he'd dated once. When she ignores his requests of help and asks him why he never called her back, he said it was because she spoke about herself in the third person.
Used in the episode "The Jimmy", where Elaine accidentally dates a guy who only refers to himself in the third person.
George's exposure to Jimmy causes him to refer to himself in the third person as well. When he realizes he's doing it, he stops, but lets it slip out several times throughout the series at moments when "George is gettin' upset!"
Bob in Becker. Lampshaded (as if it wasn't obvious enough) in one episode: Linda refers to herself in the third person and, when called on it, says, "What? Bob's the only one who's allowed to do it?"
Mendol Ikemen has Hinata, who constantly refers to herself as "Hinata" rather than "I."
Chris Rock's "Terry Armstrong" routine from Bring The Pain.
Exasperated Interviewer: You can't go through life not using the word "I"! Terry Armstrong: Terry Armstrong is missing your point. Exasperated Interviewer: What did you say at your wedding? Terry Armstrong: Terry Armstrong do.
In Season 5 of Project Runway, Suede referred to Suede quite a bit in third-person. Whether this is charming or profoundly irritating is open to debate.
In the season 2 episode of Dead Like Me called "Hurry", George works with a guy named Ted who incessantly refers to himself as "the Ted", "Tedmeister", and "Tedster".
Season 9 auditions for American Idol have introduced an odd character who spouts his nickname, "Skii Bo Ski" (emphasis on the second syllable), much the way Denny Crane spouts his name. Time will tell if he'll make it long enough for the public to vote on him.
In The X-Files episode "Duane Barry", the character Duane Barry often refers to himself in the third person.
Weaver: Does Dai speak of himself in the third person now? Dai: Dai does!
Being Human: Tully, lampshaded later in the episode when George tries on the device and is promptly mocked by his housemates for being so impressionable.
Boardwalk Empire features the bootlegger George Remus, who constantly refers to himself in the third person. "Remus finds you petty and resentful." This annoys most of the people Remus deals with, as well as being a source of hilarity, not least when Al Capone can't figure out whether Remus talking about himself or another person. And turns into a Crowning Moment of Funny when Remus is busted, and is trying to insist that Remus's bribes should keep him safe: "Remus has paid! Remus kept receipts!" Truth in Television: The real Remus was known for doing this.
"Chris Keller feels like Chris Keller's dates sometimes."
In Arrow, Oliver Queen refers to himself as "Oliver Queen" when referring to his civilian persona.
Diggle: Do you think you can do it? Ollie: The Arrow can't... but Oliver Queen can. Diggle: I've been meaning to tell you, it weirds me out to no end when you refer to yourself in the third person like that.
In Brooklyn Nine-Nine, Terry often refers to himself in the third person whenever he mentions his favourite things (or even just things he likes), usually phrased as "Terry loves [X]".