The scene with Team Dark on the beach. Despite Mephiles' attempts to turn Shadow against his friends, Shadow holds no ill will towards Omega, and Rouge comments that she'll always stand by his side. For a group of loners they really are tight knit.
Sonic leaving Elise.
Sonic's murder. Sure, it may be Narm to some, but just the very fact our childhood hero was killed...
It's more depressing when the other characters find out that the deceased blue hedgehog in front of them. A sad theme-tuned version of "His World" starts playing as Sonic's friends and allies, even both Shadow and Dr. Eggman mourn.
Amy's reaction is a particular gut punch. Starts with her calling Sonic's name in disbelief (which cracks towards the end), after which she numbly falls to her knees, before burying her face in her hands. If she's crying or not is up in the air due to a lack of graphics and the absence of sound, so all you can see is her just shuddering violently. It's much more effective than screaming and blubbering; that's anguish right there.
It's quite clear that because of her royal duties Elise has had very few friends in her life- if any at all- so seeing her only friend die in front of her is a bit much, but then she realizes that the only way to set things right is to lose the one person who likes her for who she is, not what. And she's spent a better part of her life trying not to cry to uphold her father's last wish, and in that last scene she's crying and Lacy Chabert actually does a convincing job that any player will feel bad for her.
The final scene of Silver's story. Silver and Blaze have defeated Iblis and Silver attempts to seal Iblis inside himself to prevent it from resurrecting itself once again. Unfortunately, he's rejected by Iblis. Blaze offers herself as a the vessel, since her soul is already filled with flames. As she does so she commands Silver to seal her in another dimension using Chaos Control. He refuses to, not wanting to lose her, and the two share a touching moment before Blaze floats into the sky, as her body disappears. Before she does a final blast blows away the volcanic soot-clouds allowing sunshine into the world for the first time in 200 years. Silver merely stares up at the sky as the credits begin rolling. Combine that with the awesome song that plays during the credits and you've got yourself a really sad ending. It's made even worse by the fact that Blaze is the only character who doesn't appear in the Final Story.
Sonic '06 had a LOT of parallels between stories about friendship/abandonment. As stated above, there's Elise gaining a friend in Sonic, then having to lose him for the fate of the world. There's Silver, who has to watch Blaze, his only friend who he spent his entire journey die/disappear/travel-to-another-dimension for the good of the world, and then there's Shadow, who finds out the whole world will eventually turn against him, and the person responsible for his capture was none other one of his only two friends E-123 Omega. At least Rouge tells him she'll never turn against him even if the world does... A lot of a good a dead friend does 200 years in the future though.