One interesting twist happened in "Partners in Crime". Donna and the Doctor see each other for the first time through windows on opposite sides of a room that they are both spying on. Cue completely silently mimed shouts of "DONNA!" and "DOCTOR!"
The Master demands this of the Doctor when the two of them are fighting. "Use my name!" "...Master." Repeated later: "Doctor." "Master." "I like it when you use my name."
Although it is not the character's name (who cares what WOTAN thinks?), the use of the phrase "Doctor who?" has been treated as a running gag for 50 years, and intersects with this trope in "The Bells of Saint John" when the Doctor asks Clara to keep repeating "Doctor who?" because he realizes he likes the sound of it.
Lookout: This house is surrounded. I must ask that no one leave the room. I'm Chief Superintendent Lookout. Lady Velloper: Lookout! Lookout: What? Where? Oh, me - Lookout. Lookout of the Yard. Lady Velloper: Why, what would we see? Lookout: I'm sorry? Lady Velloper: What would we see if we look out of the yard?
"The Semaphore Version of Wuthering Heights" is full of saying and screaming names, despite having no spoken lines.
In the Fourze portion of Movie War Megamax, Gentaro screams, "NADESHIKO!" as Kannagi kills her. He spends the next couple minutes softly whispering her name.
The X-Files: Agents Mulder and Scully would say each other's names in poignant moments. The names were said in various tones of voice and various volumes — sometimes they whispered, sometimes they screamed. It was nearly always surname. Scully tried "Fox" (Mulder's somewhat embarrassing first name) only twice, and he called her "Dana" only is season 1, and it kind of felt like he was deliberately messing with her.
McCloud: Chief Peter B. Clifford whenever New Mexico lawman Sam McCloud gets on his nerves, which is all the time. McCLOUD!
When Turk and J.D. reunite after Turk's honeymoon, they spend a good few minutes of saying "TURK!", "J.D.!" as they run toward each other.
Stargate SG-1: Conversations between the scientist (Daniel Jackson) and the "dumb" military commander of the team have devolved into merely repeating each other's names with various intonations: "Jack." "Daniel." "Jaaaack." "Daaaniel."
Sherlock Holmes: In the 1980s Granada production of "The Final Problem", Watson, knowing that Holmes has gone to meet Moriarty and probably his own death, repeatedly calls his friend's name over the roaring of Reichenbach Falls.
In a film adaptation of The Odyssey, Odysseus calls out to Poseidon several times after he decimates his crew in revenge for blinding the Cyclops.
The Tudors: Since nobody (barring Thomas More, his pseudo-father figure) calls Henry by his name directly to his face, for obvious reasons, the show has treated us to more than one somewhat narmful shot of people desperately shouting "Majesty!" as they are dragged off to the Tower of London or just snubbed by the king.
Wil Wheaton plays an antagonistic version of himself. Sheldon used to adore Wheaton's character Wesley on Star Trek: The Next Generation, but he became his mortal enemy when he failed to come to a fans' convention. They keep meeting, and when Sheldon is driven over the edge, he would scream in a hammy voice: "WHEEATOOON!" Then he would swear revenge. In one episode, he does this while looking toward the sky.
In The Twilight Zone episode "Nervous Man in a Four Dollar Room", the main character (Jackie) receives a phone call from George. We know it's George because Jackie says his name 16 times in the call that lasts less than a minute. When George shows up in the hotel room, Jackie calls George by his name 24 times. When George leaves, Jackie says his name another six times while he's all alone in the room, then when George shows up again, Jackie says his name another five times. This gives a grand total of 51 times that George's name is said in a 22-minute episode.
In Over There, episode 3, Hassan and his sister Sawa performed this trope when they met each other.
In A Study In Pink, John shouts Sherlock's name when he realises that Sherlock is in another building with the murderer. It's muffled through a window, which lessens the impact, but the fact that there's a dramatic echo (as well as an appropriate choice in soundtrack) helps.
John does this again in The Reichenbach Fall just before Sherlock jumps off the building, this time clearly audible.
In The Empty Hearse, Sherlock and Mary cry John's name several times each while rescuing John from inside a bonfire.
In the series 3 finale of The Inspector Lynley Mysteries, Lynley's beloved partner Barbara Havers is shot. All Lynley can manage is a frantic shout of "Havers!!!" and then a hoarse whisper of "Barbara? Barbara???"
Dexter screams Trinity's name. He is his Serial Killer nemesis and manages to escape him again: "ARTHUUURRR!"
From the third season: "MIGUEEEL!" Complete with knocking over his own computer and furniture in his office, Percussive Therapy style. It was all in Dexter's head, but he was absolutely infuriated. He thought Miguel Prado was his friend. Miguel was the first man with whom he could be open since his foster father Harry died. Said friend utterly screwed him over.
M*A*S*H: In "Operation Noselift", Hawkeye can't say more than a sentence to or about his friend Stosh without saying his name.
Horatio does say "Archie" a lot, but Archie Kennedy is his dearest friend and Heterosexual Life Partner, and moreover a broken cutie. One fan webpage kept counting the scores of how many times he says his name per episode. Episode "The Duchess and the Devil", when Archie felt Driven to Suicide and decided to starve himself, had 21 instances.
"The Examination for Lieutenant": Horatio screams Bunting's name when he's trying to desert. The honourable Horatio wants to save him and and bring him to justice, which the surly seaman of the week doesn't appreciate and considers it Cruel Mercy. Horatio's BUUN-TIIING in Slow Motion is rather laughable, though.
Data gets a couple of these. Usually when he's assumed dead, like in the episode "The Most Toys", by The Stoic Worf of all people.
In "Tin Man", it's Picard who screams. It was Patrick Stewart's line, so the odds of it being done badly were pretty slim.
Barney & Friends: BAAARNEEYY! - the kids greet their (imaginary?) friend this way almost every episode. Followed by crowding around and hugging him. One way to kick off the episode has one or more children sitting around and making Barney come to them before depressedly saying, "hello, barney", prompting the dinosaur to ask what's wrong.
In the first Mr. Show episode, during the in-universe Biopic of Ronnie Dobbs, after Ronnie dies, Movie!Terry does this while holding the corpse of Ronnie. The in-universe "real" Terry claims he did this for hours, setting a new world record for screaming.
Used at the end of a Law & Order original flavor episode in which a husband confessed to the murder his wife probably committed while he was testifying in her defense. Right after she was acquitted, the police swooped into the courtroom to arrest him, and the couple said each other's names as they were being separated. It was tragic at first. Then they just kept doing it, louder and louder, and it became something else entirely.
Interestingly enough, this scene takes the opposite approach from most other examples on this page. Instead of screaming it, Declan almost whispers it. Not out of anger, but out of fear. Because Declan realizes that he is not facing Walter White the pain-in-the-ass cook of Gus Fring's operation, but instead Heisenberg, the new drug kingpin. Not a man to be angry at, but to be afraid of.
During an Africa special on Top Gear, the boys end up stealing bits of each others cars when James' skid plate comes loose. It keeps going on, until Jezza steals a part of James' bonnet to mend his car. James is furious.
Jezza:(narrating) As dawn broke, the peace and serenity of this beautiful Ugandan morning was shattered by the bellow of a wild animal. May:CLARKSOOOOOOOOOON!!!
Clarkson does it twice earlier in the special before May does it to him. He does it first to Hammond after the latter makes them stop at a dingy hotel, and is disgusted to find skid marks on the bed sheets: "HAMMOOOND!", and then "MAY!" when he learns May stole part of his door to replace his broken skid plate.