A comedy bit where two characters, usually lovers, have to end their telephone conversation and they end up affectionately arguing over which of them should hang up first. Usually only one half of the conversation is heard and it goes somewhat like this:
Character: No, you hang up first. [a beat] Character: No, you. [a beat] Character: Okay, on three: one, two, three... [a beat] Character: No, I didn't hang up, either.
Anyone else present, likely including someone wanting to actually use the phone, will be annoyed out of their minds, especially if this becomes an Overly-Long Gag. Eventually they will usually grab the phone off the person and hang it up themselves with a innocent "Oh sorry. I thought you were talking to me". Unfortunately, all too often Truth in Television.
An example of how Love Makes You Crazy. Sickeningly Sweethearts are prone to this.
In a TV ad for the city of Las Vegas, a man and a woman go through the routine, each at their respective house, saying "Bye" to each other two or three times without hanging up. In the next scene, the man can only say "Bye" once, because she replies and then hangs up, revealing that this time she's with friends at a Vegas hotel, and wastes no time going out to enjoy the city. The man just stares at the phone, bewildered.
Mash did one in the episode "There is Nothing Like a Nurse" between (who else?) Frank Burns and Margaret Houlihan. This version had a surprise ending:
Margaret: Hang up, Frank. Frank: You hang up first. Margaret: Frank, that's so high school. Frank: Well, let's count to three and then we'll hang up together. Margaret: Okay, you start. Frank: One. Margaret: Two. Hawkeye and Trapper: [on another line] Three! Margaret: Get them Frank!
Burns then chases Pierce and Trapper out, falls in a hole, then they (Pierce and Trapper) park a Jeep on top of him
30 Rock did one between Jack Donaghy and (of all people) an unseen Maureen Dowd.
In an episode of Friends, Ross does this with Julie. Eventually Rachel (who's only recently realised she has feelings for Ross) snaps and hangs up for him... at which point Ross actually calls Julie back. "Sorry, where were we? No, YOU hang up first!"
Malcolm in the Middle inverted this in both common participants and what is argued over - Francis is angrily arguing with his mother, with both them claiming they'll hang up first. As Francis decided to repeat his claim, his mother succeeds which he acknowledges with an angry "Damn it!". Subverted in another episode where Malcolm tries to start a no-you-hang-up, but his girlfriend interrupts him by actually hanging up.
The Nanny had a variation where after a few rounds of "You Hang Up First", another character - annoyed - picked up the receiver and hung up for them.
British Sitcom My Family had Ben Harper's assistant go through this routine with her boyfriend, until he grabbed the phone from her, told the boyfriend "No, I'll hang up first." and promptly did.
Frasier has a particularly funny example of this: Niles is doing this with Maris via his cell in Frasier's living room, Frasier gets annoyed and dials a number via the house phone. Niles says "Hang on, I'm getting another call" and it turns out to be Frasier yelling "GET THE HELL OFF THE PHONE!" This, in turn, results in a damn hilarious argument over the phone from two grown men who are literally in the same room. Frasier, for his part, does come to the realization that this is ridiculous and hangs up, leading to Niles's indignant "Howdareyou hang up on me!"
It also happened in a season six episode with Ted and Zoey where the point was to show that Zoey would fight Ted about everything, including who should hang up first.
Also in the season eight episode, 'The Over-Correction', wherein Ted briefly dates a prison inmate. The usual conversation ensues, until the violent prison inmate shouts at Ted to "hang up, you little bitch."
In The Big Bang Theory episode "The Large Hadron Collision", Howard says it to his girlfriend, who promptly does so.
Haley does this with her boyfriend on Modern Family—and then is somewhat hurt when he actually does hang up.
In Hannah Montana episode, "My best friend's boyfriend", Miley's best friend, Lilly is on the phone with her boyfriend, Lucas and tells him to hang up first. "No you! No you!" Eventually Miley gets irritated and grabs the phone out of her hand and says "No, me!"
Played straight in Being Ian. The older brother does this with his girlfriend.
Subverted in The Simpsons Movie; after the man tells his girlfriend to hang up first, she says "Okay!" and immediately does so. Naturally, he's shocked and heartbroken.
Played straight at least once in the series - Grampa is telling the story of why he hates Santa, which of course starts off on an aircraft carrier in the Pacific during World War II. In the middle of a dogfight, he yells at his radio operator (Montgomery Burns) to radio a warning to another pilot, but Monty is too busy playing "you sign off first" with his girl on the other end.
Family Guy used the classic routine, word for word, in a scene with Adolf Hitler and Eva Braun, only, instead of hanging up first, they were arguing over who would take their Cyanide Pill first.
Subverted in Phineas and Ferb, since Candace's boyfriend Jeremy isn't nearly as love-crazy as she is:
Customer service representatives are usually not allowed to hang up before the customer does, but in order to stop this kind of thing from happening, they are allowed to hang up if the customer obviously has no other issues concerning the product being serviced.
This trope is a wonderful example of a deadlock, a situation which is a cause of many errors in computing: two or more programs (or threads, processes...) are waiting for each other to do something before they do anything else, so none does anything at all. If they haven't been programmed against it, the programs can literally run forever doing nothing, and this is a common cause of hang-ups.