Not a huge one, but when Amy first sees Shadow and hugs him, despite his cold personality, he doesn't seem to mind, and doesn't even use any physical force.
The Meteor Herd theme has the lines "Took a shuttle to space and left from our home, At least we're with friends and I'm not all alone." Even though it's indirect, this is the first time Knuckles has acknowledged Sonic, Tails and Amy as his friends. His theme songs from the Adventure games mostly talk about what a loner he is, so it's sweet to see that he appreciates them being there with him.
After Knuckles saves Rouge from falling to her death, not only does he apologize for hurting her, but just before she leaves, Rouge glances back at him with a smile on her face.
There's also a moment where Shadow rescues Rouge because she reminds him of Maria.
Right after Knuckles saves Rouge there is a long moment... before she angrily yanks her hand away.
In the scene before the Final Rush stage, Knuckles sees Sonic appearing right before him dropping to the ground and comes to check on him. Sonic explains that he used Chaos Control with the fake Emerald to escape the exploding capsule Eggman trapped him with. Knuckles picks him up on his shoulder, asking him if he's okay. This scene pretty much shows that despite their differences, the two do look out for each other.
In the Hero Story ending, there's Tails and Amy realizing that Sonic survived the capsule explosion, which took away the sadness the two friends had when they thought he was dead.
Amy reasoning with Shadow to save the Earth.
Amy: I know that people fight over the most trivial things. Some people may be selfish, like the professor said... but they're basically good, if they try their best and never give up on their wishes. They always have a reason to be happy; that's why you should help them out! Saving them is a good thing! Shadow, I beg you, please do it for them! Give them a chance!
Thanks to Amy's speech, Shadow finally remembers what Maria's last request really meant. Boarders into Tear Jerker territory.
A lot of people seem to think it's narm, but Maria's speech itself is heartbreaking and heartwarming all at once. She's been shot, and is dying, but says to Shadow:
Maria: Shadow, I beg of you, please do it for me... for a better future! For all people who live on that planet... give them a chance to be happy... Let them live for their dreams. Shadow, I know you can do it. That's the reason you were brought into this world. Sayonara, Shadow the Hedgehog.
During the Finalhazard fight, Shadow shows just how much respect he has for Sonic.
The entire Finalhazard battle is Heartwarming and Awesome all rolled up into one. Sonic and Shadow are constantly giving each other encouragement - they attack it in turns, and remind each other to rest and regain their Rings when it's the other's turn to fight. There are also the calls coming in from their teammates to further cheer them on (yes, even Eggman), and at one point Shadow will call out to Maria, telling her to watch him fulfilling her wish.
There's also Tails and Eggman sharing a moment:
Eggman: As a child, I looked up to my grandfather because of all the great things he accomplished in his life. He was my hero, and I wanted to be a great scientist like him. But... did he really mean to destroy us? Tails: I don't know, but what I do know is we all did it together! Eggman: You're right.
The last scene where Knuckles and Rouge share a moment, with Rouge declaring that she's thinking of something else other than jewels while holding and staring at Shadow's ring.
There's also Sonic's final farewell to Shadow. It was heartwarming considering that the two started off as rivals and then became allies in the end.
Tons of moments with Chao.
Even after death, Chao that were friends in their past lives remain friends.
Crazy Gadget in Story Mode is this as well; Amy is being held at gunpoint, Tails is powerless to stop him at this point, and Sonic has to rush on over there with the fake emerald before anything crazy happens. And throughout the level, Eggman will taunt you with the fact that he will kill Amy, with the music adding to the overall urgency. Add the very level Sonic has to go through to reach her to all this, and it'll show: do not mess with Sonic's friends, no matter what.