The intro. It's the entire relationship of the old man and his wife, from when they first meet as kids to her unfortunate infertility (why they couldn't have children) to when she dies of a heart attack, just before he's able to give her the trip to South America that she's always wanted. The creators were worried that the old man's reasons for going on his fantastic voyage would appear to be suicidal - "he just wants to join his wife".
Notice the church that Carl's sitting in? It's the same church that he and Ellie got married in.
Speaking of promises:
Russel: Cross your heart?
Carl: *pauses* ... Cross my heart.
When Dug gets called "bad dog" by someone he loves so much. Dug's heart is visibly broken.
The saga of Russell's parents, after he says that "Phyllis" told him that he was bugging his father by calling him so much..
Carl: Phyllis? You call your own mother by her first name?
Russell: Phyllis isn't my mom.
Carl: (the realization hits him) ...Oh.
Somewhere halfway between Crowning Moment of Heartwarming and Epic Tearjerker. The scene at the end, where it shows the house on the cliff... It's mighty powerful.
Many of the key Tearjerker moments can easily be attributed to Michael Giacchino's wonderful score during those dialogue-less moments. Hell, just listening to certain tracks ("Married Life", "The Ellie Badge" and "Things We Did" for those following along at home) makes those tears well up all over again.
The house seemed so insignificant when he first gets it there, but when it lands there by chance, it was as big as it was in the picture that Ellie had made as a child. This might even be a metaphor for Carl's heart, as it seemed to grow since the first time the house landed.
Another one that can be interpreted as heartwarming and sad at the same time: "Russell, for assisting the elderly, and for performing above and beyond the call of duty, I would like to award you the highest honor I can bestow: The Ellie Badge." After watching the movie, we all know how much this means for both Russell and Carl.
Speaking of Russell, his story of his father and him would sit at the corner and count cars while eating ice cream ending it with "I know that sounds dull, but the dull moments are the ones I miss the most."
"Thanks for the adventure! Now go have another one."
If you can see those nine words in context without bawling like a baby (or at a bare minimum Manly Tears), you have a heart of stone.
Dug being thrown from the cliff by Alpha and whimpering in pain.
Kevin calling desperately to her babies after being injured, and making it just a few steps towards them before collapsing.
When Carl accidentally hits the construction worker with his walking frame; you can tell by the look on his face he didn't want to hurt the guy who was actually being somewhat nice to him.
The look on Dug's face when he says "I do not like the cone of shame".
Carl: Of course you can, you're my dog, aren't you? And I'm your master!
Dug: You are my master?! Oh boy, oh boy!
"I was hiding under your porch because I love you!"
To see Carl, after repetitively telling Dug "I'M NOT YOUR MASTER!" and even insulting him unfairly earlier on, accept and give back the retriever's affection (and giving us the final proof he's finally moving on), is enough to make person weep with happiness.
Dug's line immediately afterwards is made of this too. He just gets the happiest look he's ever had (which is quite a feat, since "happy" is his default expression) and joyfully cries out "YOU'RE MY MASTER?!" before jumping up and licking Carl's face.
Listen to Muntz's theme song from the credits, and keep in mind Carl and Ellie's relationship. Then try not to hear it as a love song from Carl to Ellie: "My spirit of adventure is you..." *sniffle* Again, a very good reason the score won the Oscar...
The fact that Carl kept 'The Ellie Badge' after all those years.
Muntz falling to his death in the climax. For every foul thing he did throughout the whole film, it still stings a bit to see the brief but haunting look of sheer, unwavering fear on his face after the ropes attaching him to the house snap, and he plummets down and out of sight from several thousand feet up in the air. You can actually see that's he's still reaching for Carl right as he starts falling, albeit now it looks like he's reaching for help... Carl himself pauses for a bit when he sees it happen; it's a real Alas, Poor Villain moment for all it's worth.
The Reveal showing "The Spirit of Adventure" script on the nose of the Zeppelin.
A meta example is how a 10-year-old cancer patient had very much wanted to see the film, but was bed-ridden and hooked up to life support. Her mother contacted Pixar, and Pixar sent representatives to screen a DVD copy of the film for the girl, and give her some movie tie-in merchandise for free. At that point, the girl was unable to open her eyes, and so her mother described every scene to her as she watched the movie. She passed away several hours afterwards - but she left happy, having seen Up.
What's so awesome about this was how Pixar took no credit for this; they didn't hype it up as "Hey, look, we're helping this girl." They did it simply because it was the right thing to do.