Episodes 12 and 13. Nagasumi finally confirms his devotion to Sun by refusing to marry Lunar despite the fact that there's a serious danger of getting himself killed by her father if he refuses. And this is before he knows that Sun is coming for him. Speaking of which: Sun attempting to cut her ties with her family and fight her way through a bunch of Yakuza thugs to get to Nagasumi.
The Seto Gang gets one as well when. Despite the fact Sun publicly cuts her ties with the gang, they all stand up and get ready to rumble. Why? Because she reminds them so much of their boss.
Lunar in these episodes, particularly her interactions with Nagasumi, her remorse for hurting Sun, and her ultimate I Want My Beloved to Be Happy resolution at the end. The manga version of the story has her actually vocalize in her thoughts that she can't be angry at Nagasumi and Sun for loving each other and that she just wants to stay a part of their life since she cares about them both.
The school trip: Akeno offers to give Nagasumi a special arrangement where he'll be left alone by the mermaids in exchange for breaking things off with Sun. Keep in mind that the whole reason the two of them were betrothed is because one of them would have to die if they didn't get married. Akeno gives him a chance to escape his predicament... and he refuses. It's the point where it becomes absolutely unambiguous that he's in the relationship because he loves Sun rather than because of the threat of death.
Episode 24: Pretty much the entire cast's response to learning (however incorrectly) that Kai is dying. While most of them spend the bulk of the series butting heads with him, the moment they hear the news they all go out of their way to make his last days as enjoyable as comfortable. Even Nagasumi, who basically thinks Kai is a total jerk, treats him with respect, and everyone is genuinely saddened by the prospect of his death. Even if they do spend a lot of time fighting with each other, it slots their relationships firmly in Vitriolic Best Buds territory. Sure, it all resets at the end of the episode, but it's nice to know that the cast do really care about him.
It's also rather heartwarming to see Kai finally (though temporarily) overcome his phobia of open spaces, as well as confront his issues of inadequacy. When everyone confirms that, yes, they do see him as a friend, he seems genuinely at peace with his fate.
Chimp's reaction is both this and a genuine Tear Jerker. He proves that it's not just Kai's money that keeps him at his side and that he truly cares for him to the point where he would trade places with the allegedly dying Kai if he could.
The finale: Nagasumi storms a Mermaid Noble's castle, crashing through the walls with Mikawa's submarine and picking a fight with a very power mermaid solely because he wants to protect Sun. His desire to see her safe is so great that it allows him to spontaneously manifest superpowers.
As a bonus, Lunar and Kai both show that for all their bluster, they're more than willing to give Nagasumi all the help they can in assisting his attempt to rescue Sun.
Similarly, it's a blink-and-you'll-miss-it moment, but right before Nagasmui goes to confront Yoshio, Sun's father tosses him a coat and declares him a part of the Seto family before telling him to go rescue Sun. Not only is this the first time in the entire series that Gozaburo show's Nagasumi any form of approval, but he's intensely overprotective of Sun, and the fact that he tells Nagasumi to save her rather than going after her himself shows that he actually trusts Nagasumi enough to trust him with his daughter. For all his animosity and vitriol towards Nagasumi during the series, he really recognizes that Nagasumi's love for Sun is absolutely genuine.
Sun manages to shake off a paralyzing incense because Nagasumi proclaims his love for her. The incense shouldn't even let her hear his words, but she manages it anyways.
A somewhat roundabout example, but when Nagasumi drinks a potion that makes him appear to women as whatever they find to be sexy, Ren basically sees him as a mish-mash of himself with a lot of Gozaburo thrown in. For all that he annoys her, she really does love him. Compare this with Nagasumi's own mother, who sees Nagasumi exactly as Masa.
The fact a bunch of hardened Yakuza genuinely enjoyed two little girls singing in Lunar's flashback. They may be their bosses daughters, but still, there was no indication any of them were faking the enthusiasm.
Listen to Sun's performance in the subbed version. She's much more hesitant and, in all honesty, not that good. It still manages to make several yakuza shed Manly Tears.