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, which is probably just as melancholy as the ending. It may not be as heartbreaking as the beginning scene from Up, but good god is it tearful. Not only are we shown the Andy we know playing with his toys through an old betamax tape which constantly reminds us this is all in the past, not only do we see him aging over time, or that we see that his entire toy family is decreasing, BUT this is all accompanied with the original "You've Got A Friend In Me". It turns right into Tear Jerker territory when it ends on the line Our friendship will never die...
Arguably made even more heartbreaking in some of the translations. For instance, the Latin Spanish translation replaces that line with "El tiempo no volará" (literally Time will not fly)...
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.
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.
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.
When Bonnie first met Jessie, the score 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.
When Andy's dog comes onto the screen, all old and sleepy whereas in the previous film he ran about like a madman. Oh Pixar, you broke the hearts of college kids the world over.
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.
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 lead up to this movie, it shows that Woody is still Andy's favorite toy.
Lotso's fate. Some see it as a Fate Worse than Death. However, the Garbage man took Lotso out of nostalgia.
During Ken's attempt to help the toys, Lotso simply tells him "She's a Barbie doll, Ken! There's a hundred million others just like her!". And despite all wishes to the contrary, he's right; Ken could find another relatively easy. But Ken says 'Screw that' with one simple line:
Ken: (looking at Barbie) Not to me!
Even brainwashed, Buzz still has strong feelings for Jessie. "Bewitching good looks," anyone?
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.
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 unnoticable 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.