Star Trek found new life on the big screen when J.J. Abrams brought the franchise back with a new timeline in 2009. That film's success spawned a pair of sequels and we were expecting a fourth film to come along in short order. However, the most recent word is that the film is not currently an active concern for the studio. Having said that, Zachary Quinto certainly hopes that he gets a chance to play Spock again at some point.
I don’t know what’s happening with that, and I’m trying to figure out what, if anything, will be the future for it. I feel glad that the franchise is having its own life continuing on [with Star Trek: Discovery] and my hope is that we’ll also get to go back and play those roles again. But there’s no telling right now if it’ll happen or not.
Almost as soon as Star Trek Beyond hit theaters we were expecting a fourth film to get the green light. Franchise producer J.J. Abrams was already talking about an idea that would have brought Chris Hemsworth, who had a cameo in the first film as the father of James T. Kirk, back into the series, and everybody seemed very excited. Director S.J. Clarkson was brought on to direct the film and everything appeared to be going smoothly.
However, it seems that because Star Trek Beyond had a somewhat lackluster box office, Paramount Pictures was looking to save money and wanted Chrises Pine and Hemsworth to take pay cuts for the project. Negotiations stalled out, and from there what had seemed to be a promising new movie appears to have fallen apart.
Chris Hemsworth has recently stated that, in the end, he didn't think there was enough of a reason to move forward with the film, and so he decided to walk away from it.
Star Trek as a franchise certainly isn't going anywhere. As Zachary Quinto mentions, Star Trek: Discovery, which includes his character Spock, though played by a different actor, is still keeping the popular brand going. It seems almost certain that we will get more Star Trek on the big screen at some point, but we simply have no idea what form that will take.
Quentin Tarantino has mentioned an idea for a Star Trek movie that looks like it has as good a shot as anything of getting made, though it's unclear if his idea would include the current movie cast or not.