While trapped aboard the derelict space station in "Fear Factory", the gang is led into an illusion of Lighting Lad's room. The second Lighting Lad identifies the location, Superman, Brainiac 5 and Saturn Girl all take up defensive positions to guard and protect him.
The end of "Chained Lightning", when Lightning Lad and Lord are reunited with their little sister. D'AWWW!
The episode "Who Am I" had many Superman/Chameleon Boy moments, the twist? Some of the most heartwarming scenes were between Chameleon Boy and Ron-Karr.
The turn out by literally every member of the Legion for Ferro Lad's funeral at the end of the first season. He may not have thought much of himself, he may not have believed he was worthy of being a Legionnaire, but he knew what it took to be a hero, and everyone at in the Legion whose lives were touched by him turned out to honor his sacrifice. The eulogy Superman gave also counts.
Superman: Heroes have strength, heroes have courage, but most of all, heroes put others before themselves. Ferro Lad was a Legionnaire for only a short time, but he was a great hero, and a great friend. He gave his life to save the galaxy, and we won't ever forget his sacrifice. Long Live Ferro Lad. Long Live the Legion!
Legionnaires (fists raised high): Long Live the Legion!
Camera pans upward to reveal a massive statue in Ferro Lad's honor in the Hall of Fallen Legionnaires.
Maybe not as powerful as the above examples, but there is one in the episode "Timber Wolf" when Timber Wolf in his monster form (the Legion doesn't know he is, or was human at the time) protects an unconscious Saturn Girl, brought her to a cave then dragged her closer to a warming fire he built like a cat would a kitten.
Also counts as a Heartwarming in Hindsight when after he's restored to normal (well as close to normal as he can get) and after they've confronted his father the Legion pretty much takes him in and becomes his new family.
Dr. Londo: But for how long? A day? A week? You'll never be able to control yourself Brin.
Saturn Girl: Yes, he will (puts hand on Brin's chest and he smiles) and the Legion will be there to help. (Lightning Lad, Brainiac Five and Bouncing Boy smile at Brin in the background)
In "Cry Wolf" Phantom Girl and Chameleon Boy stay by Brin's side even though he's being hunted by the Legion and Science Police Officers, convicted of killing his father and even he's not sure whether or not he killed him. When he has a freak out and reverts into his primal form after he discovers that he was the real killer Phantom Girl tries to comfort him by saying that he's not a monster. Near the Episode's conclusion Timber Wolf and Phantom Girl share a hug after he refuses to kill his father (or rather his clone)
Lightning Lad visiting Matter-Eater Lad and Saturn Girl at the hospital during "In The Beginning," with Garth even making sure Tenzil is eating properly so he can get better.
"Dark Victory" ends with Triplicate Girl reunited with her third self and Superman X admitting that he was closer to the Legion than he admitted, complete with donning a flight ring and reciting the Legion of Super Heroes oath before returning to his proper time.