Juliet dies shortly after being caught, much to Sawyer's dismay. As a result, he refuses to be used in the next gym. Ruby states he wasn't going to use him anyway, as it was the electric gym. After he loses his pokemon but ultimately wins the gym, he finds Sawyer eagerly awaiting him outside. Their faces are too much.
From the epilogue, Giovanni deciding to turn himself in, after seeing a picture of Ruby and his Sceptile.
Despite his reluctant pick, Ruby picking Oshawott in Unova. Just look at the little otter's face!
Ruby makes it clear that he didn't want Luke on his team and that he's only keeping him on his team until he found a better Pokémon to replace him with. By the time they reach the first gym, Ruby warmed up to Oshawott and considers him an invaluable addition to the party.
Ruby warming up to certain Pokémon seems to be a recurring theme in the series. It started with Wishmur, then Spheal, all Pokémon that were in the box by the time of the Fire Red finale and Luke.
Spuds and Barb adopt a Togepi egg at long last, since Nidoqueen's inability to breed meant they couldn't have kids of their own.
Pretty much everything involving Spuds and Barb, save for maybe some of their more shallow interactions early on. Special note goes to the end of the Christmas Episode and their aforementioned egg adoption.
Roxanne visiting May after their battle resulted in Marie and Claire's deaths, and the exchange between them that helps May come to terms with what happened. Given how gym leaders who kill somebody's starterare generally depicted, the compassion and caring she shows is really refreshing to see, as is May admitting her own arrogance played a role.
Sparradile's Nuzlocke Challenge:
Pepsi, after being raised into a Nidoking, finally has to admit to Chan that he's not poisonous. Chan replies that he's never been anything but helpful to the party, and all this means is that she can hug him without risk of being poisoned by his spikes.
Paratroop's Nuzlocke Challenge:
Paratroop writes a postcard to their mother, who has difficulty with going outside, and mentions a gift they left for her in the garden. the gift is a picture of Para and their parents at a picnic signed, "Remember us being outside, I missed that-P"
Kynim's Nuzlocke Runs:
YMMV, but Fantina's "shape up" lecture to Kynim after the latter's pyrrhic victory at Hearthome Gym, along with providing a doll to symbolically bury Tyrus, was quite touching in a tough-love kind of way.
In White version, Elesa taking Bianca to talk to Social Services about her father's abusive behavior.
Jordan and Mimi's blossoming romance can almost always lead to a big case of d'aww.