The Fudomine team's backstories, full of despair and sadness until Tachibana came in. Made even more of a tearjerker when we learn Tachibana's own bittersweet story that leads him towards the team: how his pride and his violent tendences almost ruined Chitose's career and he couldn't forgive himself for years
Shuusuke and Yuuta's backstory is pretty sad as well. Specially when we see that Fuji always loved and cared for the little brother who thought he was The Unfavorite, but poor Yuuta was just too confused and depressed to appreciate it.
Eiji. Oishi. Eiji having an Heroic BSOD when Oishi sacrifices his spot in the team to let Tezuka return. And the anime makes it even more heartwretching as Eiji cries and screams and reminds Oishi of The Promise they made, grabbing him by the shirt and shaking him, going all "how dare you do this to me!? How dare you!"
Another Oishi-centered one is when he reveals to Momoshiro that, right before the Hyoutei matches, he severely injured his arms. He had refused to tell the others (only mentioning that he saved a pregnant woman from Death by Falling Over), but then Momo learns the truth, and his eyes go wide as he sees that Oishi isstill smiling to the crying mother-to-be before telling Momo to take his place... While the scene isn't overmotional, Oishi's selflesness just tugs at your heartstrings once you realize how hard must this be for him...
But one that should really pull your strings is the Anime's season finale, where Ryoma has a match with Tezuka so he can end their rivalry and move mentally to the US open. The match is intense, and then you wanna cry near the end, cause Ryoma's crying too. What's more, Ryoma wins by using all the signature techniques of his teammates, saying thank you to each of them when he uses their move. Oh yeah, and Ryoma ends it by finally saying "Thanks for everything...".