Modern Star Trek without Chris Pine is like the Federation without the Enterprise — empty, flat and directionless. Captain James T. Kirk has been an integral part of the franchise since day one of its JJ Abrams reinvention, a role only Pine filled since William Shatner's last round in the part in 1994's Star Trek Generations.
Sadly, it seems Pine is no longer attached to the part; the studio recently lost the Wonder Woman actor and his onscreen father Chris Hemsworth in a series of pay disputes, and will likely not return unless those issues are resolved.
Hemsworth's character is already dead, so his loss doesn't pose as much of a problem, but Pine's certainly does. If Spock is the brains behind the Enterprise, Kirk is the heart and soul of the starship, then how can the franchise possibly hope to continue without its captain? In the original series, movies and animated show, Jim Kirk and his crew spent many stardates going on adventures together, allowing other characters to be built to roles that might have dealt with this sort of loss. Pine's Kirk, however, has only been in three movies, and losing him at this stage in the Kelvin timeline is premature. His James Kirk still has a long way to go, and Zachary Quinto's Spock likely cannot possibly hold the fort, and the fandom, on his own.
In theory, however, moving on without Chris Pine is possible. Star Trek has persisted without Kirk before, and even flourished for decades without its decorated hero. Assuming Kirk isn't recast, there are other ways the franchise could survive.
Limit The Scope
Most Star Trek films are larger than life, often involving the entire known universe. Everyone's fates hang in the balance, and it's up to the Enterprise to keep the peace and restore status quo. In Kirk's absence, this would be tricky to pull off. Inevitably, writers would have to explain where he is, what he is doing, and why he isn't jumping in to help save the day. The first two Abrams-era films involved the greater portion of the known universe, and if the studio decides to retain Spock (basically Kirk's other half) and tell a storyline of this magnitude, the resulting film would feel mismatched, awkward and imbalanced, its holes too difficult to plug.
However, the reboots could take a page from Gene Roddenberry's original Trek and create something more compressed and episodic, involving a lot less people and requiring only a handful of characters to take the lead. This way, Kirk's absence could be more easily justified. Perhaps the Enterprise is sent on a covert mission involving only Spock and a portion of the crew. They could even be on a different starship, while Kirk remains on the Enterprise. Think Rogue One, Fantastic Beasts, and Ant-Man and The Wasp. Side-stories and spinoffs. Any event that could exist on the periphery, or occur during in-betweens, would probably work.