Tohru Honda is responsible for several of these: At the end of Volume 6, where she chases after Kyou, who has turned into a giant, foul-smelling monster. Tohru pleads for him to come back home, saying that she wants to be with him. Kyou turns back in to a human, hugs Tohru and turns in to a cat. Tohru then carries him home.
Another Tohru example: At the end of the manga, Akito has a knife and has cornered Tohru at the edge of a cliff. Tohru, instead of freaking out, asks Akito if she wishes to be friends with her. This would be a Crowning Moment Of Awesome, except the purpose was for it to be heartwarming.
Kyo and Yuki retrieving Tohru from her relatives-from-hell... Also the scene where they race back to the house after realizing they're leaving her alone for New Years.
When Kisa, the Cute Mute, comes to bring Tohru home from work, running up to her calling her "Onee-san". Tohru broke down in tears.
There's also the part where shy little Machi Kuragi says she ran all over the school to just say 'hello' to Yuki.
Again with Machi, at the end of the manga, after the curse is broken, when Yuki hugs Machi for the first time. There's then a heartwarming Say My Name moment between the too.
During Hatori's flashbacks of Kana, one of the things that leads fans to 'aww' in the middle of the sadness is Kana's declaration that she doesn't feel any differently about him because of his curse.
Hell, the entire last chapter is made up of Crowning Moment of Heartwarming and Tear Jerker amoments. From "You sure picked an odd place to get embarrassed" to "This is the key to my new apartment. You can come by any time" [...] * pretends to throw key out window* "YOU LIAR" and "When you go, will I have to come with you?" * blush* "You cunning child".
When Kagura confesses to Kyo that her love for him had only been because she was looking down on him; Kyo rejects her feelings but adds that, regardless of the reason, he was happy when Kagura played with him. Kagura then cries that she still loves him, and he hugs her and says "Thank you." Very heartwarming.
In Volume 16, when Katsuya turns up at Kyoko's parents' house just as they're kicking her out and lays into them about how they're treating their daughter. And then later on in that volume, when Kyoko realises that she's been neglecting Tohru in her grief and runs home...only to find Tohru waiting for her patiently and smiling.
Ayame, normally so flamboyant, so outspoken, says sincerely and gently to Mine, when his curse has broken: "I was born to love you — and finally, finally I can tell you that!"
Related to the previous, Ayame tells Mine that he was originally instead going to tell her a secret, strongly hinting that he was going to tell her about the Sohma family curse, in the hopes that she could love him despite it. It would have been the clearest indication that he had fully matured beyond his youthful It's All About Me attitude. It became moot, however, when the curse is broken mere moments before he's about to confess to Mine.
A non-Tohru example, toward the end of the manga when the curse is finally broken, and "God," surrounded by the spirits of all the animals from the banquet (including the cat) turns and says thank you and goodbye.
At the end of the montage showing the zodiac curse being broken, there's also a panel showing God and the cat reconciling. After learning of their history, and how two close friends broke apart because God didn't want to hear what the cat had to tell him, it's incredibly sweet to see that in the end, they rekindled their friendship.
Toward the end of the manga, there was speculation about who would be on the final cover, just Tohru or Tohru with the Sohmas or her friends or some other combination. Turns out, it was none of those, it was Tohru's mother.
Rin immediately stepping in to defend Tohru after Kagura slaps her (for not admitting her feelings to Kyo), showing just how far Rin had come since she was first introduced.
It's anime-exclusive: near the end of the last episode, when Akito is about to hurt Tohru, ranting about how the Sohma will be happy he's gone when he dies Tohru tells him that she would be sad if he died and she believes so will the rest of the Sohma.
Hiro and his mother Satsuki gets a joint one in chapter 118 after his curse breaks. Not only is his first reaction to pick up and hug his little sister, Satsuki then hugs him and says it's okay to feel conflicted because the curse was a part of him for his whole life.
There's another one before that: Satsuki asks him if he wants to hug his sister, but he declines because he's scared of turning and hurting her. Satsuki reassures him by telling him that if he turns, she'll just hug the both of them tightly.
Kyo, who hates leeks, making Tohru leek soup when she was sick. Also anytime he lets the jerk part of his Jerk with a Heart of Gold status go and acts nice, especially when he's around Tohru who he falls in love with.
After making her dinner, he sees that she's obviously upset about something. He tells her to vent to him, if she has to. He then listens to everything upsetting her, and bluntly tells her how she shouldn't be getting so worked up over such things (she's upset about a single bad test grade and how that seems like an insult to Yuki, who helped her study for it). He gets that all through to her when no one else could, and she's cheered up enough to get over her illness.
In Volume 22, after Tohru gets out of hospital, Kyo has to chase her down to explain how confused his was about her love confession and confesses to her and says how they want to be together. Tohru asks if they can hold hands and they already are. Then he asks if he can hug her. They do. Then there's a Beat and they realise he hasn't transformed.
At the very end hearing their grandchild talk about how "lovey-dovey" they are together and seeing the elderly couple walk hand in hand.
"You did good."
Kyo's obsession with beating Yuki is typically hostile, but he expresses blatant worry at times when Yuki is vulnerable. When Yuki collapses from an asthma attack Kyo abates his fight with Haru to see if he's ok, and when Yuki is confronted by Akito Kyo can be seen watching from afar with a frown on his face. This shows that while Kyo's resentment and jealousy drive their relationship, he does care for Yuki's well-being deep down, or at least does not want to see Yuki hurt by anyone other than himself.
When a young Kyou and Kazuma were walking in the neighbourhood together, other Sohmas started bad-mouthing them. Kazuma's reaction? Lifting Kyou on his shoulders.
"From now on, you will have to breathe the air of higher places."
Also, later on, when he calls Kyou "his son". Almost his entire life, Kyou had been conflicted about whether he was worthy to call Kazuma his father.
Tohru's grandfather, whenever he's not a Cloud Cuckoo Lander. Chronologically, first by his immediate acceptance of Kyouko, who was shunned by the rest of the family. He takes care of Tohru while Kyouko grieves and after Kyouko's death and also visits her grave. Furthermore, it is hinted that his weirder tendencies such as calling Tohru Kyouko-san are just an act. Kyou calls him out on this being in very poor taste, but then her grandfather says that in that way, he kept Kyouko alive for Tohru.
When Kyouko is first getting to know Katsuya, he mentions he resents his father, who was very strict and expected a lot of him; after Tohru's birth, Kyouko and Katsuya bring the baby to see his dad. Kyouko is just telling him that she doesn't think Katsuya really hates him or holds anything against him... when Katsuya mentions he wants to go get the baby's coat for her, and almost casually sets little Tohru down in his father's lap. Awwww.
Despite having been rivals their entire lives and possibly pitted against each other by their mothers, Machi and Kakeru seem to actually get along quite well. Particularly evident when Machi brings Tohru a present, and Yuki asks her if that's something she likes, because Kakeru had told him earlier that if he doesn't know what another person likes, he just buys them something he'd like. When she asks him how he knew, he just replies, "Like brother, like sister."
Furthermore, Yuki's kindness towards Machi. The very fact that he asks her what her opinion is on things even when they're trivial like what colour she'd like for her pen.
Why hasn't anyone mentioned the ending of the manga? Spoilers here but to me it's one of the most romantic and at the same time saddest moments in anime and manga history. The way Kyo realizes that he can protect and love Tohru and the fact that he himself was loved was just great and how both Kyo and Tohru reconcile and become a couple is really, really sweet. Lets grow old together indeed...
And there's Kyo's thoughts: "It's taken me a long time to fulfill my promise, but now I'll keep it for the rest of our lives."
Meshou is normally a crying, shivering Fragile Flower, but the scene in which she fully expresses in words how much she cares and love for her son Ritsu is made of this.
When the group is on vacation, Akito makes an unexpected arrival while Yuki is on a walk. Tohru worries that the two will run into each other and something bad might happen to Yuki, so she goes out to find him. He's perfectly fine and happily greets her, walking back to the house with her. And we then see that he did meet Akito, who engaged in her usual verbal abuse of him. This time though, he was able to shake it off and hold on to his new resolution, to open up his tightly-concealed emotions. The resolution itself is also sweet, since he's spent his entire life believing that there's nothing interesting about him. He finally believes otherwise, and will prove it.
While the story ends sadly, the beginning of the origins of the zodiac are quite heartwarming. God was all alone and sad, until one little cat came to be his friend. And from there, he was befriended by more and more animals, who all stayed with him and had many good times.
Kakeru might be annoying, but he obviously goes far to look after anyone he cares about. He's adorably devoted to his girlfriend, tries his hardest to help Machi, and is willing to listen to Yuki's problems (even if his advice is not always helpful). It explains why he's so thrilled at the thought of Machi and Yuki hooking up, especially with Yuki being so nice to his little sister.
Ayame trying to patch up his relationship with Yuki. Even though they go through several ups and downs, Ayame desperately wants to be the big brother he should have been when they were kids.