From the games, any Pokémon that evolves by friendship. After taking care of it for so long, it likes you enough to evolve from pure happiness.
An especially noteworthy example is Golbat into Crobat. Keep in mind that their lowest form, Zubat, is one of the original Goddamn Bats, not to mention it's not too terribly strong at first. But if you manage to raise one for yourself, it goes from one of the most annoying Pokémon of all time to one of the most reliable sweepers in the series.
Take a max-happiness Pokémon to Dr. Footstep and he'll tell you your 'mon's innermost feelings about you. They're usually quite touching and cute, especially if it's a "scary" Pokémon (like a Haunter or a Darkrai), or a "slow" Pokémon (like a Bibarel) that's grown sufficiently fond of you.
Most Pokémon: "<Name> is...a remarkable human and Trainer. That I can always perform the best any <Pokémon species> possibly can...I attribute that entirely to my partner <name>. When we travel, I can see wild Pokémon eying us enviously..." Scary Pokémon: "There are no Pokémon that dislike humans... Only humans that dislike Pokémon... We <Pokémon species> are especially shunned... But <name> always treats me as a friend and partner..." Slow Pokémon: "Hnurrr... Me? Uhm... Other Pokémon, they say to me... "You're not thinking, are you?" How insulting they should say so. It isn't true what they say. Think, think, think, I do that. All the time, I think hard how to help <name>. I concentrate! Hnurrr..."
Other Pokémon: "If I said there can be friendship between Pokémon and people, will <Pokémon species>'s friends understand? But if you see <Name> and <Pokémon species>, you will understand. Because <name> and <Pokémon species> are friends!"
This especially applies to Legendaries, since many of them have a Base Happiness of 0 (which basically means they hate you from the moment they're captured). You have to put in extra effort to max out their happiness, but seeing a Pokémon like Giratina or Darkrai give you nothing but the highest praise is VERY much worth it.
The players mother encouraging him/her at the start of the journey.
Volume Five of Pokémon: Diamond and Pearl Adventure! has Cyrus manipulating Hareta to the extent that one of Hareta's friends (a pawn in the plan) tries to kill herself. When Cyrus gives his evil tirade, he explains that the only true emotions of humanity are "hatred, jealousy, and burning anger", and gloats when Hareta unleashes fury against Saturn's Rhyperior. He caps off his speech with "You must hate me even more now." Hareta's reaction? SOBBING. He tells Cyrus "You do bad things, but I don't hate you," and talks about how he still wants them to battle someday. Cyrus is in disbelief and is nearly overcome, starting to save Hareta from the explosions (the building is blowing up around them them the whole time). Saturn stops him from going back in, but Cyrus leaves a Master Ball for Hareta just the same. Later, it's revealed that even though he got to the Spear Pillar first, Cyrus waited several hours for Hareta to show up. That reawakened human side is key in Cyrus's redemption in later volumes.
The Tepig◊ card shows a young boy holding his Tepig while posing for a family photo with his mother and father. The Pignite◊ card shows that same happy family after a Time Skip (and the addition of a little sister). Finally, the Emboar◊ card shows another Time Skip, and now the boy is married with a kid of his own. All three cards emphasize how much the Pokémon really is part of the family.
Some extra Team Rocket Heartwarming (they sure do it a lot): In one episode, James nearly gets himself killed by trying to capture a rare and powerful Pokémon on his own, and while Jessie comes to rescue him, they both end up getting trapped by the raging Pokémon. Resigned to the fact that they're going to die, Jessie tells James that, if there's an afterlife, she wants to meet him again there. This actually makes the terrified James smile, knowing that he and his best friend won't separate even in death. Luckily, they end up getting saved by Meowth, but still.
Even sweeter because Meowth and Jessie, discouraged by their bad luck, had just abandoned James to go pursue careers. He was extremely upset, but as soon as they found out he was in trouble they ran to rescue him. Then they teamed up again. Status Quo Is God, but still, awww.
Anything where Pikachu almost dies and Ash saves him and they end up hugging.
Meta example from Bulbapedia: "Today is Monday February 27, 2012 - Anniversary edition: Welcome to the World of Pokémon - In celebration of the anniversary, all Generation I Pokémon species articles will feature their artwork from Red and Green Versions." To all of those of us tropers who grew up in The '90s, the sheer amount of Nostalgia Filter cranks the "heartwarming" part up several notches.
On a video of "Emotion" from B/W, a YouTube commentator (by the username JTpiano24) commented about what Pokemon taught him. He said that Red taught him to not fear adventure, Gold taught him to not be afraid of change, Ruby taught him to not tamper with nature and let it run its course, Diamond taught him to not use religion to rule over others, and lastly, B/W taught him that beings had emotions, and that people needed to accept others to achieve peace.
This comic titled "It's The Thought That Counts which is about a Quilladin who has a crush on a female Braixen. The idea of this sounds cute, but each attempt to charm her doesn't go well for him. But the real heartwarming moment of the comic is how it ends.