Every time you rescue a sage, you get transported to a place similar to the Temple of Light from Ocarina of Time, and you receive words of encouragement from the rescued sage. Each time you exit through the portal, Link raises his sword and the sages cheer for him. Once you've rescued all of them, the Triforce of Courage appears and you're congratulated by each of the sages one by one as you climb the stairs to the Triforce piece.
Most of the sages (namely, Gulley, Osfala, Seres, Oren, Irene, and Rosso) have people in Hyrule who are worried about them (respectively: Gulley's parents; Sahasrahla; the Priest and Dampé; the friendly Zoras; the Witch; and the Craftsman); after you rescue each one, you can tell these people that their loved one is safe, much to their relief and/or gratitude. There's no in-game reward for doing so, but it feels good to put their minds at ease.
Zelda stating that she isn't very different at all from Hilda. She really felt deep sympathy towards Hilda and her being driven to desperation to save her people.
Link and Zelda's wish to help Lorule. Considering all of the trouble they were put through because of this other world, it's really sweet to see them help the parallel kingdom.
On a similar note, Hilda breaking into tears of gratitude after Lorule's Triforce is restored thanks to Link and Zelda's wish.
It's a small thing, but whenever you load a save file or enter a dungeon, the loading screen shows the Triforce. If you're in Hyrule, it's the one iconic to the series, bright and gold. If you're in Lorule, it's their upside-down Triforce, but it's dark and grey. After you've been playing for a while, you'll become used to both images. As the end credits start, both Triforces appear on opposite sides of the game's title, and just as the series' fanfare begins to play, Lorule's Triforce suddenly changes from dark grey to bright gold, shining just as brilliantly as its opposite.
Seeing Ravio added to the legend mural in Hyrule Castle was the icing on the cake after the events mentioned above. He may not think much of himself for running away, but he clearly did so much behind the scenes, helping Link save both kingdoms. It's sweet to see his role recognized in the new legend in the end.
Forging the level 3 Master Sword has its own heartwarming moment. The Lorule Blacksmith feels alive for the first time in so long and thanks you for the job. His wife also tells you how his passion for his work is what drew her to him in the first place and apologizes for being cold with you before.
Also, speaking to the Hyrule Blacksmith with the level 3 Master Sword has him completely astonished that there is someone out there so much better than him... and decides now he has a new goal as a Blacksmith, to be good enough to do that kind of work.