A huge one is when Maurice is reunited with the other servants. It's extremely difficult to watch that scene without crying tears of joy.
Jacqueline cleaning the cuts on Danielle's back after the latter was whipped, while agreeing that Marguerite deserved the beating she got.
It takes a while to notice, but after Marguerite burns Danielle's book, Danielle meets Prince Henry again, who tells her she left her copy of "Utopia" in the carriage by accident yesterday. The book Marguerite burned was actually from the Prince's library. In a sense, Prince Henry has saved a piece of her father for Danielle.
The prince came to rescue Danielle, after he had horribly rejected and humilated her at the ball. To his surprise, she had already rescued herself. He then apologized to her, admitting that he was wrong and he betrayed her trust. But it turns out that she had already forgiven him for it.
Prince Henry (in apologizing for being a Jerkass) spontaneously calls Danielle by her real name (instead of Nicole, her mother's name, which he has been using up this point). She begs him to repeat not the apology, but the part where he says her name.
Another Heartwarming Moment when he finally proposes to her.
"I kneel before you not as a prince, but as a man in love... But I would feel like a king if you, Danielle de Barbarac, would be my wife."
While it doubles as a Funny Moment because of its over-the-top nature, the wedding scene between Henry and the Spanish Princess Gabriela. The poor girl is sobbing uncontrollably, begging her parents to let her marry her lover (a Spanish courtier). Henry is kneeling at the altar seriously, and then, all of a sudden, bursts out laughing at the absurdity of it all, and despite the language barrier, he calls off the wedding and lets her go to her beloved.
On top of that, he tells her, "I know exactly how you feel," before giving her a friendly kiss on the cheek.
Some found it heartwarming that Jacqueline was able to find love at the ball (she and Laurent hit it off), even after her mother spitefully told her, "You're only going for the food." This, along with tending to Danielle's wounds and a thousand other little slights she's borne, led to Jacqueline finally deciding that she deserved better than being The Unfavorite, prompting her to turn against her mother.
Da Vinci encouraging Danielle to go to the ball and tell Henry the truth wins Heartwarming points:
Danielle: A bird may love a fish, signore, but where will they live?