After the battle with the Thunder Brothers, Inyuasha sees Kagome and Shippo convinced they're now spirits on their way to the afterlife he rushes up to Kagome and begs her not to go. Thankfully the flames were just some protective foxfire from the fur pelt belonging to Shippo's father. Once he discovers this Inuyasha denies everything, but it's far too late.
"InuYasha, I love you. I love you as a half-demon!"
Another one occurs when Miroku and Sango encounter Kagura inside Mt. Hakurei and Sango is knocked unconscious. In order to protect Sango, Miroku baffles Kagura by using his kazaana in spite of the multitudes of poisonous insects which were supposed to prevent him from doing so, severely poisoning himself before escaping carrying Sango over his shoulder, and shielding her with his own body when he collapses.
Miroku: "Heh, who do you think I am!? If in exchange, this beloved girl can live longer, then I don't need my life!"
This is followed soon after by a scene in which Miroku, having collapsed from all the poison he took in while protecting Sango, asks her to continue without him and survive. Sango vehemently refuses, throwing herself over his body and declaring tearfully that she'd rather stay and die with him.
This troper always teared up during the end of the first season opening of InuYasha (shown on youtube) when Inuyasha is walking all alone in the forest, lonely, and he then sees the full-grown Kirara coming toward him with all his friends on her back. Kagome reaches down and picks him up. They all smile down at him as if saying 'You'll never be alone anymore!' and he smiles back, realizing this. Aww!
Sesshomaru's remorse over getting Rin killed when traveling to Hell and his inability to save her. This is basically the pinnacle of human emotion from the most stoic character in the series. It's not so bad that he cries, because he never would, but seeing him get even this remorseful is nothing short of amazing. It gets even better when she's brought back.
Bankotsu and Jakotsu drinking sake on the porch as the sun sets. Jakotsu hands over the sacred jewel shards he's collected, never even considering the possibility of keeping them for himself as the rest of the Shichinintai seems to be doing, and Bankotsu is visibly touched, embracing him and declaring him "the only one in the world I can really rely on." It's the last time they ever see one another.
Awwww. I just teared up.
Kikyo giving up on the easy way to defeat Naraku to save Kohaku's life permanently.
During the "demon rats" storyline, when Sango is tearing herself apart blaming herself for not being able to kill Kohaku while Naraku is using him as a puppet, Miroku places his hand over hers and gently reassures her:
Miroku: It's all right. That's what it means to have a human heart.
The whole last episode.
Sesshomaru's Pet the Dog episode with Rin. Very early in the series, Sesshomaru is recovering from his fight with Inuyasha and is surprised a little girl is bringing him food and is smiling to see him. We're shown that she's an orphan and is abused by the villagers for stealing food to give to Sesshomaru. She's eventually killed when Koga's wolves wipe the village out. When Sesshomaru goes on his way, he finds Rin's lifeless body on the ground and while saying that it was a test for Tensaiga, obviously was repaying her kindness to him. Rin became his Morality Pet afterwords.
When Inuyasha first learns to master the Wind Scar/Kaze No Kizu, he uses it in defense against Sesshomaru, resulting in Sesshomaru being sent flying several miles away with grievous injuries. Afterwards, Toutousai points out that Inuyasha hadn't swinged Tessaiga with all his force, meaning he hadn't been willing to kill his older brother, no matter how much he might hate him. Inuyasha, of course, denies this.
When studying for her exams back at home, Kagome falls asleep on her books. She wakes up to find blanket around her and Inuyasha sitting on her bed. She starts to cry in joy, but he freaks out thinking he did something wrong. When she says it's because she's happy to see him, he doesn't seem to know how to reply, glancing around and giving a weak , "Ah...okay."
The few times it's revealed that Inuyasha and Koga deep down actually care about each other, such as when Koga might defend Inuyasha in a fight (while claiming it was accidental) or when Inuyasha reveals his human form for a second to protect Koga, and on another occasion is relieved when Koga turns out to be alive after his fight with Ginkotsu.
Also when Mt. Hakurei (which was used as a protective home for Naraku while developing a stronger body) collapses, Inuyasha and co. have managed to escape when Kagome realizes that Koga is still trapped inside and will end up being absorbed by Naraku. Inuyasha proceeds to return to save him, without Kagome even asking.
In the anime, Rin wants to stay together with Sesshoumaru, but eventually realizes that her lifespan will be much shorter than his and asks him if, after she has died, he'll remember her. His reply is a stoic, "Such a foolish thing."
So, so much of "The Tragic Love Song of Destiny", but one moment stands out in particular. Early on in their relationship, Kikyo secretly prepares to fit Inuyasha with the same beads of subjugation Kagome will use in the future. However, when they meet the next day, he first gives her a small shell of lipstick his mother left him before she died. Kikyo is so profoundly moved by this gesture that she can't go through with it, pretending to have forgotten the "gift" she promised him and electing instead to trust him.
One story has Inu-Yasha looking for a powerful shield-casting demon, which he must kill to give his sword the power to break Naraku's protective barriers. He discovers that the demon is actually a little girl who's half-demon, and who's being forced to protect her demon grandfather to protect her mother. While her grandfather thinks humans are worthless, though, it's revealed that his son, the girl's father, really did love her mother and had threatened to leave his clan if the other demons didn't honor his wishes and leave his wife's village alone. At the end of the story, Inu-Yasha turns down the chance to use the girl to power his sword, insisting that he'll just find an even stronger shield demon. The girl, as thanks for helping her, gives him a jewel that powered her abilities. Breaking it gives his sword the power it needed.