Dori, Nori and Ori had a sister. She died when they were fleeing Smaug, and Dori is the only one who remembers her.
Bilbo's description of why he didn't kill Sméagol.
Bilbo: I had never seen anyone look so utterly miserable. Here he was, this hobbit who once had been much like me, alone under his mountain, forced out of the sun, with no one but his precious to talk to, and I had stolen it from him. I could not kill him. How do you kill someone when you've just broken their heart?
Dori's description of market day in Erebor, especially when he mentions a baker who made a type of sweet bread, and Thorin thinks he remembers that baker.
There had been one baker who had always made sure to slip them such a treat, though. Thorin had liked him for not making a public show of their interest in his goods as so many others had. When Dís had read their mother's account of what had become of the dwarf, many years later, she had cried bitterly.
Fíli recites one of his grandmother's records of a Dwarf who died after Smaug came. Fíli himself is crying by the time he finishes.
Fíli:[reciting] When the dwarves moved on the next morning, despite her pleas, Mavol and his friends remained behind. What became of them we do not know. They have passed out of this world and into memory and it is to memory I commend them. Remember them, people of Erebor, for they are your heroes. They gave their lives that we might live and so we honour them. When, in days to come, there are dwarves in Erebor once more, their names will be given to the stone that they should never be forgotten. Mavol, Galgin, Davil, Ruon, Rula, Mera, Nalir. I, Dílna, wife of Thráin, Princess of Erebor, swear that you will be known to all who come after you.
Balin talks about Bifur after his injury.
Balin: I remember seeing Bifur one day, after he had recovered from the wound, sat in the same place he had always chosen when he was whittling. He had been forbidden the mines at that point because he still had bad days and found it hard to concentrate for extended periods. He sat for a long time with a finished toy in hand and none of the children were brave enough to come and get it. The axe frightened them and Bifur could not speak to them to reassure them. In the end he left it on the ground and one of them picked it up after he was gone.
Shall History Remember?:
Frodo accidentally kills a Rohirric soldier. He's heartbroken about it afterwards.
Sigrid sees Bofur be carried away by orcs. She has to choose between going after him and putting everyone else in danger, or leaving him. She chooses to leave him.
Her heart shattered.
They rode away.
Bofur and Gandalf's conversation when both are prisoners.
Bofur: Has [Saruman] lost his mind?
Gandalf: Yes, Bofur. I rather believe he has. [...] He was a great man once. This, this goes against everything we were sent to do. I do not understand how he came to lose his way so thoroughly, but he is Saruman the White no longer.
Sméagol's death. Especially his last moments, as he hopes that he has led the Nazgûl far enough away from his friends, and wonders if his sacrifice will atone for everything he did when he had the Ring.
Merry and Pippin grieving him.
Pippin: We aren't meant to be two anymore. There should be three. But there isn't and... and there isn't ever going to be three again, is there?
Fíli and Kíli realising they'll most likely die in Mordor.
Kíli's death shortly after that realisation.
At least there's one moment that will bring Tears of Joy: he didn't die.
Everyone's reactions when Mount Doom erupts, Dís's and Thranduil's most of all.