(written from a Production point of view)
On 18 July 2016, it was announced that J.D. Payne and Patrick McKay would return to write the fourth film in the current series, and that the plot would have James T. Kirk crossing paths with his father George, played again by Chris Hemsworth, reprising the role from Star Trek. 
After the death of Pavel Chekov actor Anton Yelchin, Simon Pegg commented, "It's very difficult to talk about the future at the moment because there would be such a gaping hole in it all." He personally does not want to see the role of Pavel Chekov recast.  J.J. Abrams concurred, "There's no recasting. I can't possibly imagine that, and I think Anton deserves better." When asked if they would kill off the character, he replied he had thought about it, but it was too soon to discuss.  The remaining main cast was expected to return.
Kirk actor Chris Pine has stated that, for the fourth film in the current series, he would like it to be "a slower film." Pine added, "That'd be kind of fun. Kirk and team land on a planet and go explore. It's just not going to happen. But it would be fun to make the Merchant Ivory version, a slow, talky film." When asked if he would be interested in doing a time travel comedy similar to Star Trek IV: The Voyage Home, he responded, "Pushing the boundaries would certainly be fun and I love doing comedy. If Simon's behind it... The most fun I have on set is laughing so anytime we can do that I'm way, way into it.'" (SFX, issue 276, p. 55)
It was indicated the film would be released in 2019. 
In April 2018, Zachary Quinto commented that he believed that there were at least three scripts in development for the film, with the Beyond sequel and Quentin Tarantino's pitch being just two of them.  In the same month, he reiterated that Paramount wanted to make at least one more Star Trek film, implying that the aforementioned scripts could all be turned into films, but Paramount had to decide on the order.  At CinemaCon 2018, Paramount CEO Jim Gianopulos revealed the studio was working on two new Star Trek films.  The following day, it was reported that S.J. Clarkson was in negotiations to direct this film, which was not Tarantino and Smith's project. Clarkson, a veteran of TV who made her feature film debut with 2010's Toast, would be the first woman to direct a Star Trek picture.  Eventually, it was actually confirmed that Clarkson would be the director of the new film. 
According to the Hollywood Reporter, the film was indefinitely delayed following a pay dispute with Pine and Hemsworth in August 2018: the actors allegedly wanted a larger salary, but Paramount refused in an attempt to keep the budget down, arguing Star Trek is not as profitable as the other franchise films the actors have starred in. It caused both actors to walk away from the project. 
While Paramount made it appear in their summer 2018 press releases that they were stymied by excessive salary demands from the principal cast, the situation was actually a lot more nuanced; Pine and Quinto had already been signed for the fourth film in June 2015, with financial compensation set,   whereas Hemsworth was signed a little over a year later in July 2016, before Beyond was even released.  Faced with adverse financial circumstances afterwards brought on by an almost unbroken string of financially disappointing film productions,   which included the poor profitability of Beyond,  and having in the intervening years lost the lucrative Marvel Comics film franchise in 2011 as well,  Paramount was in effect trying to break open the already signed contracts and renegotiate the salaries downwards, which was what the two actors did not agree to.  The August 2018 Hollywood Reporter report implied that the other primary cast members on the other hand, were open to renegotiation,  with most of them, including Quinto, remaining hopeful the film would yet be realized.   All primary cast regulars were originally signed for a three-film deal, which was concluded with Beyond.
With no progress made at the start of 2019, Clarkson's schedule was freed up to take on the pilot for a Game of Thrones prequel instead. 
On 9 January 2019, it was reported that Paramount had "shelved" – an industry euphemism for cancellation – Star Trek 4 indefinitely, reportedly for being overwhelmed by the breakout success of the in-the-meantime-revitalized Star Wars film franchise and the sky-rocketing success of the post-2011 Marvel Comics films, which had elevated Chris Hemsworth to super stardom status by then.  Two months earlier, in November 2018, Bad Robot Productions (also the co-producer of the far more successful Star Wars and Mission: Impossible film franchises) had already announced the termination of their original 2006 motion picture development deal with Paramount.  This decision was actually very much facilitated by the circumstance that Bad Robot owner Abrams had turned his back on Star Trek after Star Trek Into Darkness, because he became frustrated over legalities surrounding merchandising rights in which his company had a major financial stake,   and had greatly dialed down his personal involvement in Beyond afterwards to that of titular co-producer only. And even though the studio did not publicly conceded it as such, losing one of its most important production partners, if not the most important one where Star Trek was concerned, became a major contributing factor for their cancellation decision as well.
In the meantime though, and despite the information that has come to light since the original cancellation announcement, the franchise has stubbornly clung to its demonstrable falsehood, that the non-realization of the film was solely and exclusively due to Hemsworth and Pine refusing to be signed for the film, even as late as February 2020,  and June 2021, continuing to deflect any wrongdoings from themselves. 
Incidentally, this was a case of Star Trek history repeating itself; during the period 1988-1991, the film division of the studio found itself in near identical financial circumstances, complete with third party investors deserting the studio in droves, and which had actually led to the studio cancellation of Star Trek VI: The Undiscovered Country initially as well, before that decision was overturned by conglomerate higher-ups later on (for specifics on this particular case, see the main article).
Links and references
- Chris Pine as James T. Kirk
- Zachary Quinto as Spock
- Zoe Saldana as Nyota Uhura
- Simon Pegg as Montgomery Scott
- Karl Urban as Leonard McCoy
- John Cho as Hikaru Sulu
- Chris Hemsworth as George Kirk
- Based upon Star Trek created by Gene Roddenberry
- J.J. Abrams – Producer
- Bryan Burk – Producer
- S.J. Clarkson – Director
- David Ellison – Executive Producer
- Dana Goldberg – Executive Producer
- Tommy Harper – Executive Producer
- Alex Kurtzman – Producer
- Patrick McKay – Writer
- John D. Payne – Writer
- Mark L. Smith – Writer
- Victoria Thomas – Casting
- Lindsey Weber – Producer