In #30, when Joe, accompanying Jou Oohara's studies, realizes that the conversion into Barizorg is irrevertible and, as such, there's no way Cid is coming back. Joe's devastated reaction makes it all the more heartbreaking.
#38. Joe finally has to destroy Barizorg. Which then leads to Joe carrying on a conversation with Sid-sempai's ghost. And if all that wasn't gut-wrenching enough for you, soon after that, we cut to Marvelous, who has collapsed in the cockpit of the GokaiGalleon after the battle. Navi has a mini-freakout because he thinks that Marvelous is dead and has left him all alone. It doesn't have the same emotional weight as the previous scene, but it might be enough to be the straw that breaks the camel's tear duct.
Another potential example in the same episode: Warz Gill's reaction when he's told that Barizorg is dead. He spent every previous episode acting like a spoiled brat, but when he learns that the person he considered his only real friend is gone, he falls back into his chair and lets out a yell so pained that you can feel his anguish. For once, a villain's Roaring Rampage of Revenge is portrayed in the same light as it would be if the situation happened to one of the Rangers.
There is the added fact that this rampage disputes Joe's earlier question whether Warz Gill would be willing to die for this person.
In #40, when Domon finds Gai's letter and sees the picture with Honami, the woman he loves but he had to leave in the past, and Mirai, his son. He drops Manly Tears in a very heartwarming moment.
Pretty much all of #41 is this (combined with heartwarming), but Ahim breaking down and crying while the rest of the crew pledge their support for her prior to their fight against the monster who destroyed her home and killed her parents really takes the cake.
Though the next episode makes it slightly more heartwarming by showing Sally stuffing the bomb into her compartment, sacrificing her life to save her new friend.
Captain Marvelous actor, Ryota Ozawa, breaking down and crying during the Gokaiger final live show. Even if you don't understand what he's saying, you can tell just by watching him that he truly loved being able to work on the show. You can watch it here Obvious bonus points since Ryota's character would probably have been the last character to break down like that, yet we know Marvelous feels the same about his crew.