Chancellor Gorkon's death is surprisingly tearjerking, especially when you consider people in both Starfleet and the Klingon Empire worked TOGETHER in order to assassinate him. Because they didn't want him to make an alliance between their two governments!
His last words are heartbreaking: "Don't let it end this way, Captain..." Kirk, who expressed bigotry towards the Klingons in his own personal log due to (three of) them murdering his son, stands there shocked with cognitive dissonance: instead of cursing him in his last breath, the man was begging Kirk to save the peace process.
"Second star to the right...And straight on 'til morning." Manly Tears were shed.
After Star Trek: Generations was released, VI became doubly poignant because fans realized it really was the last time the crew would adventure together. Then there's the "Sign Off."
Merritt Butrick's cameo - the film was after he had died of AIDS complications. In-universe, Kirk unable to reconcile his death.
More like a Fridge Tearjerker, but how Valeris' betrayal affects Spock. She's second only to Saavik when it comes to Spock's proteges. He tried his best to teach her how to have hope, but she wasn't good enough to live up to it. She didn't just betray him as a Starfleet officer, she didn't just betray him as a Vulcan, she betrayed him as a surrogate daughter. That's a hard blow for anybody.
Imagine how heartrending it would have been had it been Saavik, as Meyer wanted.
The fact that among the conspirators was Admiral Cartright, a relatively minor (though friendly) character who had previously appeared in one of the more popular Star Trek films may come as a slap in the face to some fans.