Has Jack Bauer's original boss, Richard Walsh. At the end of the second episode he dies getting Jack a keycard that can help discover who The Mole is. He's barely ever mentioned again by the halfway point of season 1 and is utterly forgotten after that, but he does appear in a few of the show's Expanded Universe comics and novels set before season 1.
A few episodes later we also have Kim's friend Janet who gets suffocated to death in a hospital bed. Despite the fact that she was Kim's best friend, she's never brought up after that. Still, her death did mark one of the first real shockers in the show and is considered a real Growing the Beard moment that hammered in the fact that no one was safe.
The Adventures of Sinbad: Mustapha figured prominently into the first episode only to die in the second. He wasn't billed in the intro, though, so it was obvious from the beginning. And of course once he was dead he was basically forgotten.
This BuffySpin-Off plays a similar trick in its pilot episode, in which Angel is tasked with saving the life of a beautiful young woman. Unfortunately, she is killed by that episode's villain before they can become romantically entangled.
Then there's Doyle, who was a main character for nine episodes until his Heroic Sacrifice.
The Blacklist: Luli Zheng, one of Raymond Reddington's associate, is killed by Anslo Garrick in Episode Nine to cement his Hero Killer status. Dembe was also set up to be one soon after, but was subverted immediately after in the next episode.
Breaking Bad: Subverted with Jesse Pinkman was supposed to be killed off in Season 1, but the creator loved his character and Aaron Paul's performance so much that he changed his mind.
A more straight up example is Krazy 8 who pretty much exists to be Walt's first murder.
Combo. The viewer barely knows who he is other than he was a chubby friend of Jesse's, and his assassination deeply saddens Jesse. Jesse then gets offended when Walt asks who Combo is when he breaks down the news to Walt.
The special double-length pilot episode introduces Jesse, best friend of Xander, the show's Plucky Comic Relief. It seems that Xander and Jesse will be a regular pairing throughout the show, mirroring the Buffy and Willow friendship. Then Jesse is turned into a vampire and Xander is forced to kill him. But that's okay, because he's never mentioned again.
Joss Whedon actually planned to bring Jesse back as a ghost or form of the First Evil talking to Xander in the final season, but real-life problems prevented it. (This happened a few other times: He also had to have the late Cassie speak for the also-late Tara in one episode because Tara's actress thought having the First appear as Tara was too cruel, and then had to completely scrap an episode idea to bring Tara Back from the Dead because the actress wasn't available... among others.)
CSI: Began with a Na´ve Newcomer character who basically served to introduce the various members of the show's cast. With that out of the way, she caught two in the back of the head, turning into the second victim and confirming her status as the New Meat.
EastEnders: Reg Cox, who was actually dead before the show even opened. The first scene is them booting in his door to see where he's been all this time.
ER: Was supposed to start like this, with Julianna Margulies's character committing suicide in the pilot, but test-market audiences liked her so much that they decided to keep her in the show (she had overdosed on pills, so they just had her pull through instead of dying).
Kaylee of Firefly is set up to be this, getting shot by Dobson in the very first episode, but subverted in that she actually survives, with Mal telling Simon that she was dead serving as a psychotic (and hilarious) joke on him.
Frasier: Father Mike, a recurring character in the first two seasons, gets fired and Put on a Bus in the first episode of the third season just because the newly-arrived Kate Costas doesn't feel the need for a religious talkshow on the station, showing right away what a Bad Boss she is.
The three members of the Night's Watch (Will, Gared, and Ser Waymar Royce) who appear in the pilot. Will was a ranger who was executed for desertion by Ned Stark. Gared and Ser Waymar were killed by a White Walker while on a ranging north of the Wall.
Jory squeaked in, in just under the 5 episode limit. While significantly more badass than most examples, he still counts. His death is just a way of showing things are getting serious, without a major character being killed off.
Harper's Island: Uncle Marty plays this role as portrayer Harry Hamlin was probably the most well-known (and, unfortunately, probably the most talented) actor in the cast, and the character is dead by the end of the first episode.
A bit longer lasting than most, but Simone Deveaux basically fits this trope.
Charlie Andrews also counts. She is killed off in the episode she is introduced, then Hiro goes back in time to save her only her to reveal a dangerous deformity is already killing her. Then Hiro goes back in time again and manages to get rid of said deformity only for her to get sent back to the 1940s. (Though the fact that her actress was a main cast member on another show contributed to the last one.)
Charles Deveaux is a reversal of this trope. At the start of the series, he's already unconscious and dying; he seems little more than a plot device to pair off his daughter Simone with his nurse Peter. Almost as soon as he's actually dead, we start getting flashbacks and stories about him revealing him to be one of the chessmasters of the show.
Andy Renko (played by Charles Haid), one of the major characters, wasn't originally even supposed to make it to episode two. The plan was that at the shootout at the end of the pilot, Renko would be killed, and only his partner Hill (played by Michael Warren) would survive, to try to adjust to a new partner (Jake Mitchell, who had been in the same show with Warren before, Paris). However, in test screenings, the camaraderie between Warren and Haid was universally felt to be a special strength of the show, so Haid got a contract as a regular (and an "And X as Y" credit), and Mitchell had to settle for a single guest appearance.
Subverted Trope, where a pair of cops just happen to walk in on the wrong group of nervous drug dealers and are anticlimactically shot. The aftermath is played with appropriate drama, sending the message that, as in real life, no one is completely safe. But just as the plot of the episode has been resolved and it would be fairly easy to let it set at that, Furillo receives word that they're alive, but in critical condition....
Hustle: Did this in the premiere, by shooting the group leader through the head, and then subverted it by revealing it to be Faking the Dead.
Had Lisa Faulkner, who was sacrificed in the second episode, but shocking nonetheless. Arguably, more so. Given that she'd survived the first episode and seemed like a major character, audiences could be forgiven for thinking she'd be in the series long-term. Alas, no.
Helped by the fact that Spooks (as the show is known in the rest of the world) does not have opening or closing credits, meaning that audience has no way of knowing who's a regular cast member, who's recurring, who's a guest, etc.
Miami Vice: The show's characters were introduced in the pilot. One of them was Lt Rodriguez, who commanded the unit. Rodriguez was played by Gregory Sierra, who was a well-known actor from Barney Miller and his character appeared to be a part of the show's regular cast. Then he was killed in the fourth episode. It was Edward James Olmos who joined the regular cast as Lt Castillo, Rodriguez' replacement.
The opening episode features a sixth member of the community service group called Gary. However, he misses out on the storm that gives everyone superpowers and in the same episode gets hacked apart with an axe.
Then in series two, in a deconstruction of the Sixth Ranger trope, a character called Ollie was introduced. The promo and opening scenes of the episodes framed him as a new main character. He became part of the community service group, was given a backstory and power (teleportation). Then 15 minutes into the episode he is shot and killed by the Villain of the Week.
Gary manages to do this again in series 3 - in the Alternate Reality Episode he is shown to have survived... only to be killed by a Nazi officer testing out his new superpowers.
Oz: While the pilot introduces the show's characters, the central figure of this episode is Dino Ortolani (played by Jon Seda) who appears to be a main character. But the pilot climaxes with his murder.
The Phone: In the pilot, the first team was introduced and competed against the second team per a normal Reality TV show until about maybe half an hour (tops) into it when they don't complete the given task and are subsequently eliminated and DROWNED.
Primeval: New World introduces Evan's best friend and Dylan's partner, Drake, who has all the makings of being the Team Dad for the show. He's killed halfway into the pilot to establish that the show is going to be much darker compared to Primeval.
Robin Hood: Killed off outlaw Roy four episodes in. Somewhat diluted by the fact that after that episode, he was never mentioned again.
Which is a lot further than the rest of Little John's gang, who simply disappeared after their first episode.
The Royal Today: This failed Brit daytime soap began with three nurses fresh from college doing their first hospital shift. The blonde one got stabbed by a crazy in episode one and bled to death, but then what did you think was gonna happen?
Shark: In the first episode somebody got fired. The only real character to die on the show wasn't until the end of the first season.
The Shield: Began with Detective Terry Crowley being assigned to an LA police department's Strike Team to secretly investigate rumours of corruption and brutality. The episode ended with the show's protagonist, Vic Mackey, finding out and blowing the back of Terry's skull off. In a break with Lamb tradition, Crowley's death came back to haunt Mackey several times in the following seven years, and the character appeared in flashbacks.
This TV series had the character of Major Charles Kowalski infected with a Goa'uld parasite in the pilot episode, and killed off in the episode immediately following it. This was done after he had played a significant role in the original movie on which the show was based. Furthermore, the real use of the character is to have a situation where the defector, Teal'c, gets to prove his worth and loyalty to the Earth authorities.
And its sister show Stargate Atlantis had Colonel Sumner, who was supposed to be the military head of Atlantis... until he's captured by the Wraith, has his life sucked out, and has to be put out of his misery by then-Major Sheppard.
Stargate SG-1 started long before that, offing the entire Five-Man Band in the first episode, in order to introduce a Five-Man Band with four slightly more interesting characters. The Red Shirts guarding the Stargate at the beginning look like they'll be important... then Apophis and the Serpent Guards slaughter all but one. She is taken, it turns out, to be considered as a new host for Amonet. When Amonet turns her down, Apophis says, "A shame," brain-fries her to death in the second of the two instances of fatal Goa'uld hand-thingy brain-fry in the entire run of the franchise, and yells "Send another!" out the door.
Kirk's BFF Gary Mitchell in the second pilot "Where No Man Has Gone Before".
Basically any Red Shirt on Star Trek can be this if they are sufficiently involved with the plot.
Later in Star Trek: Voyager the Chief Medical Officer of the Voyager is introduced as a source of tension in the ship between him and Tom Paris, only to be killed along with the rest of the Medical Staff to make way to the Emergency Medical Hologram.
Supernatural: Sam's girlfriend, Jessica, in the pilot. However, she didn't fit the trope perfectly because her death was also very much needed for the plot to continue after the pilot.
Torchwood: When this Doctor WhoSpin-Off was being developed, the crew went further than usual by casting a well-respected actress as the Sacrificial Lamb and involving her character, Suzie, heavily in pre-publicity (including a special feature in the Radio Times), to increase the shock when she bought it. The show then got more mileage from both character and casting by bringing Suzie Back from the Dead temporarily for a later episode.
The Vampire Diaries: Vicki Donovan is introduced in the pilot and then killed off in the seventh episode of the show.
Lexx: Giggerota...sort of. She died in the second movie (the first season consisted of four movies), and the first movie seemed to set her up as the Token Evil Teammate...however, it was fairly obvious that writing her into an ongoing series would have been problematic given her Chronic Backstabbing Disorder, her complete lack of loyalty to any of the crew, and the fact that none of her goals were aligned with anyone else's in the slightest.