When Harry sees Dumbledore's body on the grass next to the tower in Half-Blood Prince. It was bad enough seeing that Snape actually killed Dumbledore, but the fight scene distracts you from that until Harry pushes through the crowd around Dumbledore's body. Then there's the bit with the fake locket.
It's particularly painful seeing Harry so furious at Snape that he tries to assault him with a Cruciatus Curse...which Snape nullifies while bitching out Harry.
The scene when Slughorn finally gives Harry his memory. His sad declaration that he's not proud of what he did, and the way he tremulously asks Harry not to think too badly of him after he sees it especially.
The in scene when he and Harry went to recruit Slughorn to work at Hogwarts, Dumbledore told Harry that he wouldn't need to worry about being attacked, because "You (Harry) are with me (Dumbledore)". At the end of the book, after Dumbledore took all of the potion to get the fake Horcrux, Harry is helping Dumbledore get out of the cave, leading to this exchange:
"It's going to be all right, sir," Harry said over and over again, more worried by Dumbledore's silence than he had been by his weakened voice. "We're nearly there.... I can Apparate us both back.... Don't worry...."
"I am not worried, Harry," said Dumbledore, his voice a little stronger despite the freezing water. "I am with you."
Snape's "Don't call me coward!", especially when you realize how much he went through just to get to that moment.
Similar to above, Snape looking at Dumbledore with hatred and revulsion, seems like the betrayal but it's more hate at having to do this at all.
The awful moment when when Dumbledore is begging Snape “Please" and you later realize Dumbledore was begging Snape to kill him to spare Draco from having to do it and Dumbledore from unnecessary suffering
The way that Snape and Harry have the same words describing them when they have to harm Dumbledore in some way Harry: "Hating himself, repulsed by what he was doing, Harry forced the goblet back toward Dumbledore's mouth and tipped it, so that Dumbledore drank the remainder of the potion inside." Snape: "Snape gazed for a moment at Dumbledore, and there was revulsion and hatred etched in the harsh lines of his face."
Lupin and McGonagall's reactions to Dumbledore's death. Lupin's distraught "NO!", the only time Harry had ever seen him lose control of his emotions, and McGonagall, who is always so calm and collected, almost fainting.
The end of Voldemort's job interview (as seen in a Pensieve memory). Whatever he became, Dumbledore's line still brings to light the simple fact that Tom Riddle could have been so much more, and the sorrow that Dumbledore feels over how much of a monster his student became.
Voldemort: We have nothing more to say to each other.
Dumbledore: No, nothing. [a great sadness filled his face] The time is long gone when I could frighten you with a burning wardrobe and force you to make repayment for your crimes. But I wish I could, Tom... I wish I could...
Remus describing himself as "too old, too poor, (and) too dangerous" for Tonks. It's clear just how low the poor man's self esteem is, and also suggests that after Sirius's death he doesn't want to get close to anyone again for fear of losing them. He's lost all of his closest friends, either because they died or because they detrayed everyone.
Not a specific moment, but something about how Harry, for a limited time, came to feel a certain degree of affinity and awe about the original owner of that old book, given the mutual dislike as far as their face to face communication goes.
From the Film:
From Half-Blood Prince: the addition made after Dumbledore's death, when first McGonagall, then Luna and Neville, and eventually everyone raises their wands lit by Lumos in silent honor of their fallen headmaster.
Dear lord, the scene in Half-Blood Prince where Dumbledore leads Harry to get the Horcrux, and Harry must feed him the potion, even after he cannot tolerate it, and he gets all frail and pale.
In place of Dumbledore's funeral, the wand-lit memorial by teachers and students that disintegrates the Dark Mark is pretty moving.
The look of utter devastation on McGonagall's face... and the scene afterwards.
McGonagall: You meant a great deal to him.
Sixth movie, when Malfoy was in the Room of Requirement, trying to get the Vanishing Cabinet to work, and the finch appears to have died. On top of being shown his utterly disturbed expression, you can hear him crying and you can really feel how stressed and scared he is. Pushed even further when he's in the bathroom later, breaking down.
It's enhanced by the fact that in the book, Draco is definitely sympathetic, but still clearly not to be rooted for. The movies, however, humanize him and turn him from just another caricatured antagonist into a boy who made all the wrong choices and is lonely and afraid. One may be unable to help but see the similarities between Draco and Regulus Black, as both are forced down the path of darkness out of loyalty to their family.
In Half-Blood Prince Slughorn's story about the magical fish that Lily Evans gave him, and how, when he came downstairs and looked at it one day, it had vanished - and so he knew that she was dead, because a wizard's magic only stops when they do.
It's a double one, because up until now the audience is lead to believe that Slughorn only cared for his Slug Club members to cash in on their fame and get gifts from it. However upon learning that Lily died he was completely devastated, showing that he did care for their well being.
The scene in the movie of Half-Blood Prince where Hermione is crying in the abandoned room and Harry tries to comfort her, only to have the lovebirds interrupt and be driven off by the real lovebirds. Specifically the line "How does it feel...when you see Dean with Ginny?". Christ you can hear how heartbroken she is.
The last scene in the movie. As the Golden Trio reflects on what has happened and what will come in the future, a noise echoes through the sky. It's Fawkes, singing his song to mourn his lost master and flying away from Hogwarts for good. Cue credits and audiences blubbering.
If the events of OOTP wasn't enough, Lestrange singing "I killed Sirius Black" in the cornfield to taunt Harry and Co. will elicit both Berserker Tears and sad tears.