Lt. Commander Susan Ivanova is promoted to Commander and later to Captain. In the Distant Finale, she's a General.
Lieutenant JG David Corwin is promoted to full lieutenant.
Other characters on B5 to get promotions possibly include Captain Lochley (she is a colonel in The Lost Tales, but it's not clear if it was a promotion or just part of a rank restructuring) and Zack Allen (goes from officer to chief of security. Since Station Security is set up more like a police department than a military force, those appear to be the only two named ranks, though Zack does end up replacing Garibaldi's first right-hand man after the first one shoots Garibaldi in the back).
Band of Brothers: Happens to many characters, with the protagonist Dick Winters being the most conspicuous example: he starts the series as 2nd Lieutenant and ends it as Major, outranking the officer who trained him.
Barney Miller: In the last episode, Barney gets promoted to inspector, and uniformed officer Levitt gets promoted to detective. About halfway through the series Wojo gets promoted to Sergeant after taking the exam multiple times over the years.
The Bill: Happens to multiple characters over the course of this show.
Chuck: Major John Casey gets promoted to colonel toward the end of Season Two.
CSI: Jim Brass gets a side-ways promotion to Homicide detective from Head of the CSI Lab team after Holly Gibbs get killed at a crime scene in the first episode. For the rest of the series, he is the main cop dealing with the crime lab.
CSI NY: Danny Messer was briefly promoted to sergeant after passing the exam, but then he felt betrayed by his rookies and went back to the crime lab and his old detective rank.
Colonel Lethbridge-Stewart was promoted to Brigadier between his first and second appearances.
Corporal Benton (coincidentally?) got a promotion every time the Doctor regenerated: Corporal Benton appeared with the second Doctor, Sergeant Benton with the third, and Regimental Sergeant Major Benton with the fourth.
This second promotion, however, is extremely unusual, if not downright unrealistic. Benton is seemingly promoted from Sergeant during the Pertwee era, to Regimental Sergeant Major by his final appearance, supposedly about a year or so later. In the British Army, Regimental Sergeant Major is not actually a rank but an appointment - that is, a title held by a soldier of the rank Warrant Officer class 1 (WO 1 for short). To reach this rank, Benton would have been promoted over TWO lesser ranks - from Sergeant to Staff Sergeant, than to Warrant Officer class 2 (WO 2), then to WO 1. Although this could be used as an excuse for later appearances of UNIT being set a few years into the future of the broadcast date, as an attempt to explain why Benton has moved up three ranks...
Golden Boy: The Framing Device is that Walter Clark rises through the ranks of the NYPD and becomes police commissioner after less than ten years on the job. In the pilot he is promoted to detective due to his heroic actions during a robbery and chooses to be assigned to homicide where all the other detectives have at least ten more years seniority than he. A recurring plot point in the first season is that both Det. Don Owen and Det. Tony Arroyo are eligible for promotion but Owen has been effectively blacklisted by the senior staff for the last twelve years and he needs to close a high profile case to finally get his promotion. In the season finale Owen finally gets his promotion after he and Walter catch a Professional Killer who committed 15 murders during his career.
Gomer Pyle, U.S.M.C.: An episode revolved around Gomer and the rest of his platoon going for corporal. (Or maybe it was PSC to PFC, it's been a long time since I've seen the episode.) Gomer didn't want to at first, he "liked being the backbone." (Privates are - or were - called the Backbone of the Marine Corps.)
I Dream of Jeannie: Capts. Tony Nelson and Roger Healey were promoted to major late in the first season or early in the second.
JAG Most of the promotions on the show were shown: i.e. the character raises the right hand and takes the oath of office and the insignia is switched:
Harmon Rabb, Jr. gets promoted from Lieutenant (O-3) to Captain (O-6) through the course of the entire series.
Sarah Mackenzie gets promoted only once, from Major to Lieutenant Colonel offscreen between the fourth and fifth season.
Bud Roberts goes from Ensign (O-1) to Lieutenant Commander (O-4) in nine seasons. His wife Harriet went from an Ensign to a Lieutenant (O-3).
Petty Officer Jennifer Coates gets a promotion in the ninth season.
Recurring character Thomas Boone (Terry O'Quinn) gets promoted from Captain (O-6) to Rear Admiral (O-8), and is about to become a Vice Admiral before allegations of war crimes surfaces and cancels it.
Law & Order: Special Victims Unit: John Munch was promoted after he took the Sergeant's exam as a bar bet. He ended up acting squad commander of the SVU for half an episode when Cragen got transferred to the Chief of Detectives office as punishment for his team's tendency to be Cowboy Cops.
In Season 15, Olivia Benson takes and passes the Sergeant's exam as well.
WPC Annie Cartwright in Life On Mars gets a promotion to WDC in the second series, causing some consternation from Ray Carling because it's 1973 and sexism was still okay at the time...
The series finale of Major Dad ended with the title character learning he'd been promoted to Lieutenant Colonel.
In early seasons, Lt. Colonel Flagg from Military Intelligence was a recurring character. In the fourth season he became Colonel Flagg. This paralleled the head of 4077th going from Lt. Col. Blake to Col. Potter, because Flagg couldn't be subordinate to the CO; otherwise his orders could be easily countermanded. (Not that Col. Potter listened to him much, anyways).
Also, in the last two seasons or so Klinger is promoted from Corporal to Sergeant, and Father Mulcahey from First Lieutenant to Captain.
Detective Martinez takes the sergeant's exam and passes, so he can get Put on a Bus to another precinct. (Earlier he had been promoted from undercover officer to detective, but it was never shown or mentioned in much detail.)
Bobby Simone gets promoted to detective first class (or whatever it's called) in a big ceremony.
New Meat John Clark is promoted to detective for meritorious service; at the same time, Sipowitz is promoted to detective first class, which pisses off John Clark Sr.
Sipowitz takes the Sergeant exam; he passes and becomes the desk sergeant downstairs from the squad. In the final episode he gets promoted (ahead of others waiting) to be the squad commander, still as a Sgt. He promises that this will be his last job - he's not planning to try for another promotion or transfer.
Power Rangers S.P.D.: Has its first Rank Up at the beginning, when Sky, Bridge and Syd are promoted from being police cadets-in-training to being the department's Power Rangers (though Sky is pissed to learn that he's only the Blue Ranger, while the role of team leader and Red Ranger went to new draftee Jack instead). It then has another one in the finale when Jack decides to leave the force; Sky (having matured since his last promotion) becomes the new Red Ranger and Bridge moves up from Green to Blue. And when Bridge makes a guest appearance on Power Rangers Operation Overdrive a couple seasons later, he reveals that a third set of promotions happened offscreen: the team's commanding officer was promoted to head the entire organization, Sky replaced him as head of the Earth branch, and Bridge replaced Sky as Red Ranger.
Revolution: In episode 6, upon returning to Philadelphia with Danny, Captain Tom Neville is promoted to Major.
Rizzoli & Isles: Jane's love interest 'Casey' Jones is a Lt. Col. in the US Army. After recovering from injury and returning to active duty, he is offered a promotion to Colonel. His decision to take the promotion and stay in the army is one of the reasons he and Jane break up.
Jack O'Neill gets a double treatment: promoted from colonel to brigadier general in season 8 of Stargate SG-1, and to head of Homeworld Security in season 9. By Stargate Universe he's wearing three stars.
Jack's first act as a general? To officiate the promotion of Sam Carter from major to lieutenant colonel. She had started the series at captain before being promoted to major in the third season. Sometime between The Ark of Truth and Season 4 of Stargate Atlantis, she gets her promotion to full ("full-bird") colonel.
General Hammond was Put on a Bus in SG-1: "The New Order" by giving him a third star and sending him to the Pentagon to head up Homeworld Security.
Though it went unremarked, supporting character Dr. Janet Fraiser started out wearing captain's insignia, then later switched to the brass leaf of a major.
Though it went unremarked, odd for a main character, Julian Bashir started out with the rank insignia of a lieutenant, junior grade (one gold pip, one black pip), then at the start of season four he started wearing the insignia of a full lieutenant (two gold pips). A similar thing happened with Jadzia Dax, who started the series as a lieutenant and started wearing lieutenant commander insignia at around the same time.
Riker got a field promotion to captain when Picard was captured by the Borg. When Picard was rescued Riker got demoted back to Commander.
Wesley was promoted from boy to acting ensign to actual ensign (then dropped back to cadet when he entered the academy). He left Starfleet before graduation.
In the episode "Lower Decks", several ensigns are concerned about who will be promoted. After the most likely candidate got killed in action, her friend was promoted but he didn't feel he deserved it.
Geordi La Forge goes from lieutenant junior grade in the first season, to full lieutenant in the second season, to lieutenant commander in the third to seventh seasons, to commander in the movies. And in a flash-forward in an episode of Star Trek: Voyager he's a captain.
Worf was promoted twice, starting as Lieutenant J.G., moving up to full Lieutenant off screen between seasons 2 and 3, then promoted on screen to Lieutenant Commander in the first Next Gen movie.
Inverted in the case of Data: in the "normal" flow of time, he started out and remained a Lieutenant Commander; in the final episode flash-back time (to the first episode), he wore the rank of Lieutenant.
Star Trek: Voyager. At the beginning of Season 4, Tuvok gets a promotion from lieutenant to lieutenant commander. After being demoted to ensign in "Thirty Days", Lieutenant (Junior Grade) Tom Paris soon gets his pips back as Status Quo Is God; this is in noticable contrast to Static Character Ensign Harry Kim whose lack of promotion after seven years in the Delta Quadrant is a Running Gag among fans.
In Tour of Duty, Myron Goldman goes from 2nd lieutenant to 1st lieutenant between first and second seasons, and Staff Sergeant Anderson is promoted to Sergeant 1st Class in season 3.