Lee Garlington, who played Claire, was apparently let go because of her criticisms of Larry David's writing.
The Pilot of New Girl had Coach as the third roommate to Nick and Schmidt. However his actor Damon Wayans Jr. was already committed to another TV show on a different network, ABC's Happy Endings, so the second episode of New Girl had him replaced with Winston.
Hogan's Heroes featured a Russian POW named Vladmir in its pilot episode, though when the series was picked up, actor Leonid Kinsky refused to renew his contract, as he disliked how lightly the Nazis were portrayed; so saying, Larry Hovis as Carter was promoted to full-time regular, after initially appearing as a one-shot guest character in the pilot.
In the pilot of Eureka, the head of Global Dynamics was a character named Warren King. He was replaced by Nathan Stark (offscreen) starting in the second episode.
Star Trek: The Original Series is a complicated case, as it had two pilot episodes ("The Cage" and "Where No Man Has Gone Before"), and though they both formed part of the series' first season, neither was the first episode aired.
If "Where No Man Has Gone Before", the successful pilot, had aired as the first episode, the replacement of Dr. Mark Piper with the younger but otherwise very similar Dr. Leonard McCoy would be a straightforward example of this trope. (The wholesale replacement of the regular cast, except Spock, between the first, unsuccessful, pilot, and the second, successful, pilot is probably a different trope.)
Dr. McCoy is an interesting example in that he is basically this trope in reverse: Gene Roddenberry always wanted DeForest Kelley for the Doctor, all the way back to the first pilot, but was overruled by each pilot's director (Robert Butler favoured John Hoyt as Dr. Boyce, and James Goldstone preferred Paul Fix as Dr. Piper). Only when the show went to series could Roddenberry cast Kelley without interference.
A straighter example of this trope is the Captain's Yeoman: Colt in "The Cage" (played by Laurel Goodwin) was replaced by Smith (played by Andrea Dromm) in "Where No Man Has Gone Before". Dromm declined to continue with the series to pursue a film career, and so was replaced by Grace Lee Whitney as Yeoman Rand for "The Corbomite Maneuver" (which also introduced Bones and Uhura).
Communications Officer Alden in "Where No Man Has Gone Before" is replaced by Lieutenant Uhura in the series.
The pilot movie that launched Babylon 5 was aired separately months before the series itself began, which led to cast complications and several substitutions in the first regular episode:
The First Officer, Laurel Takashima, was replaced by Susan Ivanova.
The chief surgeon, Doctor Kyle, was replaced by Doctor Franklin.
The Resident Telepath, Lyta Alexander, was replaced by Talia Winters (who was later written out and replaced with Lyta Alexander).
In The Sarah Jane Adventures Kelsey, a female school chum character in the first episode is repalced by Clyde, a male school chum character in the second.
It helps that both are Black, snarky, initially obnoxious and the Secret Keeper.
In the original CSI: Crime Scene Investigation the Audience Surrogate character, Holly Gribbs was quickly replaced with the eerily similar Sara Sidle because she was generally hated by test audiences... the creators had anticipated as much. As a result, they decided for Holly to die (rather than recover) in the second episode.
Night Court changed defense attorneys several times, but the first time was between the pilot and second episode where Gail Strickland as Sheila Garnder was replaced with Paula Kelly as Liz Williams.
The JAGPilot Movie had Caitlin Pike as Harm's partner on the USS Seahawk investigation. In the series proper she was replaced by Meg Austin.
The two JAG episodes that act as the pilot for NCIS had a character named Vivian Blackadder on Gibbs' team. She was dropped for the series proper, so the first real episode introduced Caitlin Todd as a secret service agent who transferred to NCIS for episode 2.
It happened again for NCIS: Los Angeles - Special Agent Macey was introduced as the head of the LA branch in the two-parter pilot on NCIS. Come the series proper, Macey's been replaced by Hetty.
Juliet O'Hara was introduced in the second episode of Psych, replacing Lassiter's partner from the pilot after she was transferred due to a ban on inter-office affairs.
In the pilot of Elementary Captain Gregson has a Hispanic partner named Detective Javier Abreu who was skeptical of Holmes' deductive skills and questioned the necessity for his consultation. From the second episode on, Captain Gregson has an African American partner named Detective Marcus Bell who is skeptical of Holmes' deductive skills and questions the necessity for his consultation.