Dayu's origin in Act 25, revealed in a dream Mako sees while both are in an Ayakashi's dreamworld; not only will it tear at your heart, to see Romi Paku herself in person crying out like a helpless Yandere at the zenith of her emotions is a really, really powerful moment.
Her death presents one as well, though perhaps not as potent. As she's getting crushed by Doukoku, she lets loose the sootball she's been carrying around with her. Seeing the wickedly smiling creature just call out her name over and over again at the discarded robe makes you feel even a little bit sorry for her. There's a little bit added later when, after Doukoku comes back post-battle to pick it up, (knocking it away) the sootball is STILL calling out to her. Doukoku then steps on it. Damn.
Kotoha in Act 20. If you don't already tear up by her insistent "I'm fine!" as Takeru cradles her in his arms as she unknowingly nears death every second... well, okay, you're actually tough enough to tank that up, but still! Luckily, this is a prelude to some greatmoments in camaraderie.
The moment between Ryunosuke and his best friend in Kabuki in Act 35. The scene is utterly devoid of speech except for one line his friend said at the end; which is exactly what provided the intended impact.
"Do your best... as Shinken Yellow."note Act 41
Genta at the shrine in Act 43, on his knees begging Juzo to have some humanity. Juzo's reaction makes it all the more heartrending.
The entirety of Acts 45-47 is one big Tear Jerker after another. All the vassals being torn between continuing to follow Takeru or fulfilling their duty by serving Kaoru. Jii telling Takeru how much he cherished the time he spent raising him to be the man he is now. And then finally, when Takeru broke down and cried when Ryunosuke, Chiaki, Mako, and Kotoha came and told him that while he may not be their real lord, he is still important to them.
Especially tear jerking is Ryunosuke in this: torn between his devotion to his duties as a samurai and his friendship and loyalty to Takeru, he originally stays behind at the house, trapped in a Heroic BSOD as he tries to reconcile his conflicting feelings. It is only through the help of the fisherman/Kuroko from earlier in the series that he is able to find a solution.
The ending of the series. With Dokoku destroyed, the Shinkengers part ways as they leave Shiba Clan manor one by one, giving Takeru a tearful goodbye. By the end of the episode, the only ones left in the Shiba Clan manor are once again Takeru, Jii, and the Kuroko.
Also the end of the Final Live Tour show. Just try not choking up a little bit during Tori Matsuzaka's speech. Only he and Hiroki Aiba are able to hold back tears of their own.
During his fight with Juzo in Episode 47, Takeru comes within a hair of of falling to Gedou when he thinks that he has nothing worth living for except for fighting. It takes The Power of Friendship to bring him back from the edge.