So instead Kirk goes to fetch a remote to disable the cloak on a bunch of missiles Soran (the bad guy) was about to launch. The remote just so happens to be on a rickety bridge and, as Kirk manages to make a final act of disabling the cloaking system, the bridge collapses down a cliff, taking Kirk with it.
What Really Happened:
First of all, it's clear that Kirk was shoehorned into the film only because the suits weren't confident they could get people to watch a Kirk-less Star Trek movie (Leonard Nimoy and DeForest Kelley both refused to be in the movie, saying the crew got a perfectly good sendoff in The Undiscovered Country, a film specifically written for that purpose). Then, when the writers were sitting around brainstorming ideas for, you know, what to actually do with him, somebody said, "Why don't we kill Kirk?" (yes, that's literally what they said).
So, they brought Shatner and Kirk back to the franchise specifically to kill his ass, and thus wrote in a death for him where he... gets shot in the back by the bad guy.
They filmed it, too:
That didn't make it into the movie because test audiences felt it wasn't heroic enough. So, grossly misunderstanding that feedback, they had a rusty bridge accidentally fall on him instead. Couldn't he at least been having sex with something at the time?
Preferably not a bridge.