Pretty much ANY scene about Ellie is this, not to mention the music they play during many of those scenes.
The final shot of Carl and Ellie's house, before the credits roll.
Russell's badge ceremony. Also a Crowning Moment of Funny, thanks to Russell's been-to-the-wars-and-back appearance, a stark contrast to the other badge recipients.
Carl: I'm here for him.
One occurs in the very first few minutes when young Carl tries to retrieve his balloon in the clubhouse.
"I was hiding under your porch because I love you."
Dug: Can I stay?
Carl: Can you stay? Well, you're my dog, aren't you? And I'm your master!
Dug: You are my master?! Oh boy, oh boy!
Carl, after repeatedly telling Dug "I'M NOT YOUR MASTER!" and even insulting him unfairly earlier on, accepts and gives back the retriever's affection (and giving us the final proof he's finally moving on).
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.
Especially since this is an ironic callback to when Carl opened the door and saw Russel asking to come in, and at first said, "No." Now, at the culmination of his adventure, he says to another tag-along, in so many words, "Yes."
"I don't want your help! ...I want you safe!"
"...Master? It's all right." In that moment, Dug summed up the very essence of what it means to be a dog.
Russel talking about his dad, though it crosses into Tearjerker territory.
"...I like that curb."
Carl and Russell piloting the zeppelin with all of the dogs poking their heads out of the windows.
The credits are distilled adorable; Carl taking Russell to see Star Wars is a heartwarming passing on of experiences to Russell.
The best part, for some, was paying attention to letters that cropped up twice in one word and spotting differences. They didn't use a handwriting font, the credits were actually handwritten. It's just such a nice touch.
The tie montage at the beginning, symbolizing a lifetime spent together in happiness and love.
After you see Kevin badly injured, still calling out to her babies, and then attempting to walk feebly.
The music that goes with the movie has officially crossed into Heartwarming/Tearjerker territory.
With the bus to take Carl away to the retirement village waiting, he goes back inside to get something and... takes off in the house to begin the adventure. You'd have to be dead not to feel joy there.
Dug's first appearance is made into one retroactively with the short "Dug's Special Mission": it was his birthday, and he had wished for a new master.
How about when Russell shows Kevin to Carl? Kevin promptly picks up Russell and cuddles him to herself, rocking back and forth on the ground like she was rocking him to sleep. Or when she tosses Russell around in the air. Also, when Russell convinces Kevin that Carl is nice despite how crabby he may seem, Kevin's response is to pat Carl on the head with her beak in a show of acceptance. So cute.
Pixar manages to convey very powerful messages with very simple scenes — specifically, when Russell leaves Carl to go rescue Kevin. The old man realizes that he has a new goal, and attempts to go after him, but the balloons left don't have enough helium to lift the house. Without hesitation, he starts dumping all of his things, the ones he risked his life for before. It works: Without material luggage to keep it grounded, the house flies again!
While everything else had been dumped out and allowed to smash on the ground, Ellie's chair, and his next to it, had been carefully arranged the way they had been inside the house.
It's easy to miss, but during the scene where Alpha, Beta, and Gamma first attack Carl and Russell, look really close when Alpha tells Dug, "Well, at least you now have led us to the small mailman and the one who smells of prunes." When Alpha turns to Carl and Russell, Carl pushes Russell behind him and holds out his cane at Alpha in defense. Aww, you do care, Carl!
Muntz's theme song, which also plays during the credits. Setting aside the association with him (and how its upbeat message of not giving up on your dreams is somewhat subverted by what has happened to him), its message is the same as the movie's: to go out there and live your life, and do so to the fullest, and as long as you are with those you love, it will always be an adventure. The kicker though? The song itself already sounds like a vintage love song (complete with scratchy gramophone effects and a big metaphor for love), but when you lay it against the scrapbook photos from the credits, and think about Carl and Ellie...well, it's pretty clear she was and still is his "spirit of adventure".