Pinkie asking Harry to trust her against the magical hornets and he does. Even better: she SURVIVES the scenario.
The thing that sparks Harry shrugging off the Nightmare's possession? The fact that Twilight got hurt in the process and he started to justify it. He realizes that he would never do that in his right mind.
The reason Harry is back in Equestria for False Masks? Celestia and Luna want to give him a medal for all he's done!
The love confession in Chapter 13 can and will tug on heartstrings. Pun intended.
Chapter 16: Twilight is finally reunited with Harry and Rarity... Trixie too.
At the end of Strange Friends, Harry telling Twilight his Name.
And Twilight's friendship lesson letter-
Twilight:I learned that you can become friends with almost anypony. You may think that you’re both too different from each other to ever be friends, but sometimes, if you get to know them, you’ll find that the two of you have more in common than you’d ever believe. It takes all kinds to make a world, and there’s nothing wrong with being different, even if it means the two of you make for very strange friends.
Chapter 24 is full of heartwarming, but Luna assuring Harry that he will always be welcome in Equestria no matter what Celestia says is one moment that stands out.
The end of Ditzy Dreams from Extra Stuff. Because there will never be a time where it's not heartwarming for nice people (and ponies) to tell Harry that he's an amazing and special person.