In Episode 3, Shiro has a breakdown after realizing that her chess pieces are refusing to sacrifice themselves, rendering her usual chess tactics no longer viable and putting her chances of winning in jeopardy.
The end of Episode 8. Sora is removed from existence and makes the otherwise heartwarming ending theme very disturbing by removing him from it.
And again at the end of Episode 9 when Sora is brought back to existence, he's prepared to be punched by Shiro for leaving her alone to figure out how to win the game. What does she do instead? Cries and apologizes for not being able to win sooner.
Just read volume 6 of the light novels...
Made worse when watching the Big Damn Movie after reading this volume. You know that Riku and Schwi won't live to see Disboard find peace considering how Tet words the story from the very beginning, but you can't help but just hope... and hope after seeing them fall in love and work together with the rest of the Ghosts in bringing peace.
Also from the Movie, Corone's words on not wanting to lose her family are tearjerking because Riku and Schwi are still going to die anyways. The ending actually shows Corone crying in the cave where Riku's chessboard was, wondering why she's feeling cheated despite having the peace their entire colony has been waiting for.
This whole scene also makes Stephanie's comments on Corone in the ending Harsher in Hindsight, because Steph describes her ancestor as an amazing queen who always smiled and never cried. The fact that the only times we see Corone cry is when Riku and Schwi are leaving says how much Corone had to take on in the duo's place once they passed away. Also is Irony for Stephanie herself, since Blank push a lot of things on her without always asking for her consent.
Heck, even if we saw it coming, the entire second half of the movie with Schwi's death at Jibril's hands along with Riku passing away shortly afterward as well is heartbreaking. The original "251 second limit" from the Anime Expo trailer was actually the time limit Schwi had in passing along all her data and memories to the rest of the Ex-Machina cluster to help Riku, and the entire time, she's trying not to die and escape to be with Riku again. Jibril's nonchalant manner in killing her doesn't help at all. When her last words come to pass and the scene cuts to Riku waking up to find her gone, the despair in his eyes is obvious.
In fact, all that was left of Schwi after Jibril was done with her was her ring. In her last moments, Schwi cast a barrier so that the ring alone would be untouched, and it is the only reminder of Shuvi that Riku has left once the Ex-Machina inform him of what happened.
How Tet came to be and Riku's own death. Even though Riku had achieved his goal in outsmarting the other Races and taking the Star Grail, the fact that he sacrificed almost all of the Ex-Machina and did what he thought as "spit on Schwi's spirit" was heartbreaking. It doesn't matter whether you're watching the Dub or Sub — Riku's voice actors do a heartbreakingly amazing job in showing the despair and heartbreak Riku suffers, coming to the culmination of his begging someone to come up and be the One True God. And then Tet comes to be, smiling almost sadly as Riku realizes his imaginary playmate was real all this time, and disintegrates with what the viewer can assume to be a smile, hoping to play with Tet again in another life, with Schwi at his side.
