Almost every heartwarming moment goes for the Jelly Trio's, showing The Power of Friendship very often. The most prominent ones:
In "Swimming Buddy", Raj has to deal with his acrophobia in order to be a shark with his friends. He almost surrenders, but when he sees a marshmallow with a "Congratulations" message, he gains enough courage to face it. And succeeds.
Also, in "Marshmallow Jones", it has the near ending on the episode, when Lazlo and Clam hug a "marshmallowed" Raj, and tell him they love their "marshmallow buddy". Raj, overwhelmed with happiness, tells them he loves them too and breaks down in tears. All together now: Awww.
Also, a few moments show that Edward doesn't actually hate Lazlo and even likes him back. While Lazlo hugs him almost always, the moments when Edward smiles or returns the hug instead getting angry is very heartwarming.
Lumpus' confession to Jane. Too bad he was in drag and her boyfriend mistook the confession. Hilarity Ensues.
The Rotten Bean Scouts telling Edward he's the greatest in "O Brother Who Art Thou". Edward is even moved by this remark.
The ending of 'Tomato Paste'. Edward is about to go to Tomato Camp after being accepted, and as a gift from all the campers, he's given a picture of himself. Edward is moved to tears. which is what gets him kicked out of going to Tomato Camp.
Lazlo's devotion towards Scoutmaster Lumpus, even though the Scoutmaster doesn't even like him back.
Edward's love for his doll Veronica.
Clam willing to pretend to be Nina's younger brother in Step Clam. The episode makes it pretty clear the two get along very well, contrary to the usual antagonism between the Beans and Squirrels. The final scene in the episode also reveals that Jane Doe tried the same as Nina when she was little. Her pretend brother? Slinkman!
At the end of the Slugfest episode, Lumpus confesses that the reason he didn't want Slinkman to go to the titular event is because he didn't want to be left alone, and laments that they never do anything together. Slinkman then offers for Lumpus to come to Slugfest with him, which he happily accepts. The episode ends with the two of them having fun together at the festival.
At the end of "The Nothing Club," Lazlo thanks Edward for coming up with the idea for the titular club. Edward is genuinely touched, and actually smiles.
In "Tree Hugger", it seems as if Scoutmaster Lumpus is about to chop down the Migrating Mulberry Tree, which is a critically endangered species of plant, just for the sake of finishing his boat that he had built ever since he was a boy using one of the Scouts as a chainsaw. But, when he takes another look at the tree (of whom is apparently female), he changes his mind and chooses the Tree by dumping his boat in Leaky Lake.