When Andy ultimately decides to give his toys to Bonnie.
Even more importantly, the toys are hearing about how special they all are to him, after he called them "junk" near the beginning of the film to stop his mother from selling them.
Andy gets all choked up as he's getting into his car right after.
Add to that, he actually says "Thanks guys...", as if either he knew his toys always loved him or he was thanking them for making his wonderful childhood.
Andy realises Woody was in the box of toys being donated, and Bonnie reaches out to take him. Andy's immediate instinct is to snatch Woody back and not give him up. When this causes Bonnie's expression to drop, however, a clearly torn Andy explains to her why Woody is so important to him. Once Andy is convinced Bonnie will take care of Woody, he lets her keep him.
Seeing Andy playing with the toys with Bonnie. It's what they've wanted the whole movie. Andy is leaving, but he played with them one last time. Operation: Playtime was a success after all...
And on top of it all, Woody gives up his place in the college box, to make sure that his friends got a loving home.
When Andy's toys are facing the incinerator and they calmly accept their fate and comfort each other by joining hands, every one of them. The fact that Buzz starts it by holding Jessie's hand makes it all the sweeter.
This is a subtle moment during the incinerator scene; Mr. Potato Head holds Rex's hand after spending the entire trilogy bad-mouthing him.
The beginning, where we see Andy playing with the toys as a little boy, the way we remember from the first two films, underscored by "You've Got a Friend In Me." Doubles as a Tear Jerker, especially when the song ends on "Our friendship will never die..."
Bonnie practically defines this trope. Not only is she painstakingly adorable and hilarious in every single scene, her playtime so brilliantly mirrors that of Andy's that it's impossible not to gush at her.
This scene, during Bonnie's first playtime introduction after she drinks too much pretend coffee. Caution: EXTREME Moe Ahead, will lead to Cuteness Overload in all individuals ◊
The entire Where Are They Now end credits, where Bonnie's toys have gladly welcomed Andy's into the family and Sunnyside is made "cool and groovy!" thanks to Barbie and Ken (who knew they could be so awesome?). Even the toy soldiers get a happy ending, landing at Sunnyside. The fact it ends with everyone dancing to a Spanish version of "You've Got a Friend In Me" just makes it the perfect conclusion to one of the best trilogies of all time.
This bit of dialogue just before Buzz and Jessie's dance:
Buzz: (nervous laugh) I-I don't know what came over me.
Jessie: (dips him) Just go with it, Buzz.
The smile he gives her afterwards is just the icing on the cake.
The way Potato Head looks with disgust at Woody's dancing. Kind of brings a "Back to normal" vibe to it.
Chuckles the Clown smiling at the end. Not So Stoic for kids, and just as powerful.
Not to mention that it looks like Barbie and Ken adopted Big Baby.
Lotso's old henchtoys tag teaming to be played with by the playful and rough toddlers of the Caterpillar Room, rather than leaving a single group of toys to suffer alone, all while Barbie and Ken cheer and support them. It's rough, but by working together, they can still have fun.
Seeing Andy drive to Bonnie's house hit's the heartwarming buttons for anyone who remembers the jokes between Andy and his mother about him getting to drive the car "when he's older" from the first two movies. I guess he really is all grown up now...
At the end, Andy gives Woody one last piggyback ride, just like he used to when he was a kid and as seen from the video tape. Meaningful Echo that one.
A bit Meta, but the inclusion of a Totoro doll in some scenes, truly heartwarming, because it shows the high amount of respect that Pixar has for their predecessors.
He has a small smile or a relatively neutral expression on his face for most of the film, but in the last shot before the closing credits he finally breaks out the famous Totoro grin for the first time — and he does it again, multiple times, during the credit roll.
Bullseye trying to follow Woody when he's leaving Sunnyside only for Woody to tell him to stay because he doesn't want Bullseye to be all by himself in the attic.
When Bonnie first meets Jessie. Now, for the first time since her first owner Emily abandoned her, Jessie will have a little girl to play with her. The score for that moment could have easily been "When She Loved Me"; it probably wasn't so that Toy Story fans could see the rest of the movie, instead of solid tears.
In what's almost a Fridge moment, the condition that Andy's toys are in is a heartwarming moment. Andy has had all of these toys for at least a decade. All of the toys he still has are still in very good condition. Buzz's electronic parts all work great (and his batteries have the juice to power them), the Potato Heads still have every last one of their parts, Slinky's body isn't tangled...it's clear that Andy cared about his toys, even as he got older, and was apparently made sure to either repair them or to not damage them in the first place. Sure, a lot of his old toys are gone, but look back at the toys you played with as a kid. How many of them would still be in the sort of shape that Andy's toys are in? This boy cared about his toys.
The night after Bonnie first plays with Woody. While Woody has become the new favourite, instead of acting jealous they actually compliment him on how happy he was making her.
When Woody tries to leave, Bonnie's toys are initially hurt and worried that he didn't have fun, but when he explains that he's someone else's toy already they all band together to help him get home.
Trixie looks up Andy's home address, which is right around the block. Before Woody leaves, they ask him with concern how he "escaped" the daycare with Lotso after he mentions coming from there. Then Chuckles tells the story of how Lotso became the daycare dictator. The other toys look at Woody when he realizes his friends are imprisoned there. They tell him he can't go back because he probably won't escape a second time. Nevertheless, they support his decision to sneak back into the daycare via Bonnie's backpack.
What's even more heartwarming is that Woody is willing to give up the chance to be played with everyday so he can go to college with Andy, where he'll be completely alone with nothing to do and no one to talk to. That's when you realize Woody's bond with Andy is so much more than just a toy with his owner. It's genuine, undying love/loyalty.
Early on in the movie when Andy decides to pack up his old toys and put them in the attic, he stops with Woody and puts him in his College box. After the neglect he felt in the first movie and for the decade that led up to this movie, it shows that Woody is still Andy's favorite toy.
During Ken's attempt to help the toys, Lotso ridicules him for thinking of Barbie as anything more than a replaceable piece of plastic. And as a toy who was replaced himself, Jerkass Has a Point. But even though Ken knows there are a million Barbies, none of them are quite like this one...
Lotso: She's a Barbie doll, Ken! There's a hundred million of them just like her! Ken:(looking at Barbie) Not to me, there's not.
Barbie looks so delighted when she realises Ken thinks of her as more than another Barbie doll.
Even brainwashed, Buzz still has strong feelings for Jessie. "Bewitching good looks," anyone?
When Barbie sees the toys are being imprisoned by Lotso and his gang, Ken offers her the chance to go back to the Dream House and escape the fate of her friends. She responds by immediately calling it off with them and basically demanding to be imprisoned as well. Underneath her reputation and her time so far in the film swooning over Ken, it's clear that the other toys were family to her.
And later after converted into Spanish Mode, it's quite heartwarming to see Buzz (beside his Latin Lover-like interaction to Jessie) still treats Woody (despite brief jealousy over Jessie's attention) and others like True Companions. For example, after Slinky is kicked by Lotso, Buzz, along with Woody, grabs Slinky to prevent him from falling into the dumpster. And of course, in the climax he also joins with the others in saving Woody from said dumpster, even if they had to fall into the dumpster as well.
Spanish Buzz saving Jessie from the falling television set in the garbage truck.
The utter glee in Jessie's voice when Buzz recovers from his Disney Death, followed by her planting at least a dozen kisses on his cheek.
Lotso's ultimate fate is a tie between this and Crowning Moment of Funny: A garbage man finds him in the dump, fondly reminiscing aloud over the Lots-O-Huggin' Bear he had as a kid. Then he ties Lotso to the grille of his truck, where he's advised by the other toys the garbage man has there to keep his mouth closed. He gets his wish of being with someone who loves him, and his comeuppance, all in one.
Though definitely also tinged with Nightmare Fuel due to him being one of the several Sunnyside sentries Woody and co. have to avoid, seeing Big Baby sitting on a swing and gazing up at the moon is kind of sweet.
A meta-example: While Pixar could have gotten any actor to voice college-aged Andy, their first choice was the same actor who voiced him in the first two films (John Morris). Especially since Toy Story 2 was his last role and they had to track him down. And so is for Sid, Erik von Detten still become his voice actor, although unlike John Morris, Erik already has many other roles in-between Toy Story and Toy Story 3.
A small maybe unnoticeable one, but seeing Sid all grown up and apparently behaving normal if obnoxious garbageman when in the first film as a kid he was a total Sociopath. Looks like Woody did scare him straight and wound up changing for the better.
Freeze-Frame Bonus reveals that Woody's note to Andy says 1225 Sycamore on it - nothing else. That means Andy made up his entire spiel to Bonnie off the top of his head, based solely on how well she played with her own toys. The man is truly great with kids.
The scene where Andy's mother breaks down in tears over her son growing up. She'd been little more than a satellite character up to this point, so it's nice to see that Andy growing up impacts more than his toys.
A blink-and-you-miss-it moment but just before Andy takes his toys to Bonnie, he tells his mother that he and Molly have said goodbye about ten times. It shows that even with all the crap she gave him in the beginning of the movie, Andy does care for his sister and shows he'll miss her in college.
The final shot with the slow pan upwards that shows a series of clouds which all look the same. Anyone who has seen the first movie would recognise them as Andy's wallpaper pattern from his old house. It lends the scene a sense of new beginnings.