Any scene with Athrun and Kira re-uniting outside of the battlefield.
The scene where the Justice saves the Freedom from fire, proof that Athrun would rather defy orders than side against his friend.
Also when the Archangel and the Orb ship Kusanagi launch into space, and the Freedom reaches out towards the Justice so Athrun isn't left behind. Quite possible the greatest bro moment between two Humongous Mechas.
Cagalli and Kira actually have a few dramatic hug scenes. The strongest of which is probably when The Archangel docks in Orb for the second time, and Cagalli rushes aboard to find Kira and immediately jumps into his arms. Of course this is as far as they ever go since they're brother and sister but even so their moments are rather moving.
The Cagalli-Kira-hug heartwarming is combined with Kira-Athrun-reunion heartwarming in a particular scene after Kira and Athrun disembark from their new mechs and walk up to each other after the battle of Alaska. The situation is very awkward between the two of them, as this is the first time they meet face-to-face since they murdered each other's friends and nearly killed each other... but then Cagalli runs up from behind, hugs them both, and breaks down crying, overwhelmed with happiness at seeing both men she deeply cares for safe and, apparently, friendly.
Any of the dramatic kiss scenes. Mu and Murrue's scene (in which Mu admits he disobeyed orders and saved the Archangel from destruction simply because he loved Murrue) and Athrun and Cagalli's scene (where they vow to protect one another and finally end their Slap-Slap-Kiss with, obviously, a kiss).
In Gundam SEED, after Kira defeats Andrew Waltfeld and his lover/co-pilot Aisha, the two of them quickly jump out of their seats and into each others arms; perhaps as much heart-rending as heartwarming as the mobile suit explodes immediately afterwards, presumably killing both of them. That they wanted nothing more than to hold each other at the end, however, certainly qualifies.
Actually, Andrew survives, and explicitly says that Aisha protected him at the cost of her own life. Still counts, of course.
Lacus comforting Kira after he finds out the truth about his origin, telling him that he still has a right to exist as everyone else does. It was something that she learned from her own mother who she barely knew since her childhood. Regardless of what you think or feel about Lacus, this was one of her highest points in the series.
The 1 on 1 scene where Athrun is being promoted within ZAFT after destroying the Strike and Yzak is acting unpleasant. Instead of making it harder for him, Athrun apologizes for their lost comrades and thanks him for his service. Yzak seems to take it well and tells Athrun not get himself killed, so they can keep their rivalry.
After Kira's first major battle in the Strike, Mu is called over to the Strike as Kira is refusing to come out of it. As Mu approaches and then overrides the hatch controls, he angrily calls out Kira's name, as if he's a Sergeant Rock about to chew out the New Meat. But when he sees inside and realizes Kira is nearly catatonic due to combat stress, Mu immediately switches to a much gentler voice and Big Brother Mentor demeanor, speaking encouragingly to Kira and helping him out of the cockpit.
During his period of captivity aboard the Archangel, Dearka accidentally triggers Mirallia's Berserk Button by abrasively wondering if she's an emotional wreck because "her Natural boyfriend went and got himself killed." Following Mirallia's ensuing Freak Out (in which she tries to murder him with a knife), he realizes that she did, in fact, lose someone, and kicks himself for unwittingly taunting her about it.
The next time he sees her, he asks what her boyfriend was piloting at the time of his death. When he learns it was a Skygrasper, he concludes, with relief, that "it wasn't me." He then offers Mirallia the chance to kill him, if that's what she came to his cell to do.