George Kirk's suicide run and Amanda's death left huge lumps in my throat. No other Trek has had that effect on me, with the exception of crying over Tasha's death when I was 5.
I don't know about anybody else, but I was stirred by the 'signing off' at the end of The Undiscovered Country.
I didn't believe for a second that it was really the end. And indeed, most of the characters popped up again here and there in TNG and others. Chekov and Uhura were back as recently as a few years ago in the fan film Of Gods and Men