After Gonzo learns the truth behind his alien heritage. Never before has Gonzo seemed so happy as when he shouts, "I KNOW WHERE I COME FROM!"
All of Kermit's and Gonzo's interactions. Although he may not quite understand Gonzo, Kermit never the less does his best to help and support him throughout his identity crisis- even while the other Muppets are making fun of him.
Gonzo's about to leave with his alien family, then he realizes that he just can't leave behind his Muppet friends and the the only home he's ever known. It proves just how strong the bond of friendship of the Muppets are.
Gonzo's alien brethren must have been looking for him for a long time, to let him stay on Earth at the end just goes to show how much they must really care for the guy and his happiness. D'awww
I like how Rizzo was almost hysterically happy that Gonzo wasn't going to leave Earth. Now that I think about it, it's hard to imagine things without those two together.
Kermit's very simple reaction to Gonzo's farewell puts a lump in many a fan's throat every time.
On the audio commentary, Gonzo begins to tear up while re-watching the family-photos scene, and Rizzo hugs him. Near the end of the film, when Rizzo is saying goodbye to Gonzo, the Rizzo in the commentary leaves the room sniffling.
While she did it mostly to get the news scoop, Miss Piggy earns points for fighting the government agent in order to find Gonzo's whereabouts.
Meta example: A young fan of the Muppets with bone cancer got to work as a puppeteer for the film as her wish from the Make A Wish Foundation. Becomes a Tear Jerker since she passed away a few months later. Info here.