Despite being a pretty random, comedic show based as much on Comedic Sociopathy and not so much about being good to their fellow man, the show has managed a number of heartwarming moments between them the characters:
Carly and Spencer
Pretty much every episode that centers on the sibling bond between Carly and Spencer:
iWanna Stay With Spencer shows that despite whatever Spencer appears to be, he always has the best interests of his sister in mind.
iGot A Hot Room: After the disaster of the titular hot room, Carly and Spencer's happiness once the room is complete is palpable..
iPear Store: Spencer protects Carly from Trey's advances.
Carly and Freddie
Carly and Freddie share a simultaneously simple and complex relationship, a strong friendship, and an undercurrent of romantic tension, they have been part of heartwarming moments together:
iSpeed Date: Despite their horrible night and their horrible dates, Freddie asks Carly to share a dance with someone they don't hate, and they share a slow dance together, that visibly clears away the bad memories from the rest of the night, as they dance in each others arms, with Carly's head resting gently on Freddie's shoulder.
iSaved Your Life: Freddie risks his own life for Carly, is considered a hero and starts a relationship with Carly. That is heart-warming as it is, but Freddie giving up that relationship to avoid hurting Carly in the future is arguably the strongest moment of moral fibre on the show so far, and proves that Freddie cares deeply for Carly and their friendship beyond just his romantic attraction to her.
He does it again during iStart A Fan War (caring deeply for Carly, not saving her life), after Sam screws up and Adam walks out, Freddie steps in to try and fix it for her.
iReunite With Missy: Freddie giving up the six-month cruise around the world because he knows Missy isn't looking out for Carly's best interests, even if that means he has to keep dealing with Sam.
iOMG: Carly and Freddie working together to help Sam get together with Brad dating.
iParty With Victorious: YMMV as to whether this can be considered "heartwarming" but Freddie willing to stand up for Carly after she cried upon discovering Steven was actually cheating on her.
iOwn A Restaurant: Freddie cutely asking Carly if they were too late to love each other. Their little lunch date. Setting up a study date in the hallway. Freddie helping Carly after she got flashblasted and finally Freddie protecting Carly from Spencer's attack robot.
iGoodbye: Carly answers Freddie's question from iOwn A Restaurant by initiating a kiss with him. Freddie kisses her back, then celebrates as they leave the room. They might be apart for a while, but they won't be apart forever. Freddie and Carly finally get their happy ending, Carly is over the litany of jerks and bad relationships, and Freddie has the girl of his dreams, the one he treated so well, even when it meant he lost her for a time.
Sam and Carly
Sam and Carly fight more than Carly and Freddie have, but they always know they can depend on the other and that their friendship will survive any trial:
Sam and Carly making up at the end of iQuit iCarly.
Sam and Carly making up at the end of iDon't Want To Fight
Sam attacking the bully who shoves Carly in "iMake Sam Girlier."
In "iGot a Hot Room" when Sam comforts Carly after her room's been destroyed. Just the way she says "C'mere, kid."
Freddie and Sam
Freddie and Sam might bicker and argue, but they share moments that prove beyond doubt that they care deeply for each other, even if they don't want to admit to it publically. They have a strong friendship, and an undercurrent of romantic tension. They became a couple at the end of iLost My Mind:
iKiss: Sam apologies to Freddie for publicly announcing he's never been kissed. They two share a tentative first kiss and basically reveal their "hatred" isn't really hatred at all.
iReunite With Missy: Sam coming to Freddie for help, when Missy starts to steal Carly away from their friendship with an evil plot.
And it's even more sweet when you realize Freddie gives up a six-month cruise around the world to help her. Enemies my butt.
iMake Sam Girlier: Freddie announces that despite Sam causing him "physical and emotional pain," he thinks of himself and Sam as "good friends."
iSaved Your Life: When Sam comes running to Spencer, forgetting about their paintball war screaming "Freddie's hurt!" Perhaps the only time Sam's shown concern for Freddie's physical well-being. Along with her bacon speech at the end because she doesn't want to see either of her friends hurt.
iOMG: Freddie's speech of friendship towards Sam, to try and help her overcome what he believes is her inability to put herself out there to Brad.
iLost My Mind: At the end of the episode, where Carly asks the iCarly fans what they think of Sam liking Freddie, and Freddie says that no one has even asked HIS opinion yet. Then, while Sam is worrying that he is about to embarrass her in front of the entire Internet as payback, he kisses her.
iDate Sam And Freddie: They have two adorable kisses during the episode. They also put their arms around each other. They go on multiple dates. They share a smoothie together. Sam calls Freddie "Baby".
As Carly said, though it was extreme for Sam to sticking bees in a teacher's car, it showed that Sam cares a lot about Freddie.
iCant Take It: Sam and Freddie have two more adorable kisses. Freddie says "Hey Cuteness" to Sam and Sam replies "Lips, Please". It's quite heartwarming.
Freddie refusing to be bribed with the palladium from his mom if it meant breaking up with Sam.
Considering that bar was worth around $100,000, I'd say that's pretty sweet of him.
Freddie forgives Sam for sabotaging his chances to get into that N.E.RD Camp when he realizes how truly sorry she was and that she really does love him.
When Sam says that she no longer hits Freddie in the face. Awww.
iLove You: Even though they broke up because they don't click, constantly fought, and because their personalities don't match being in a relationship with each other, saying "I love you" in turn was sweet, and shows that even though they are broken up and can't love each other in relationship, that they do love each other as friends.
Sam and Spencer
iGoodbye. The entire everything that they did together in the episode.
in iCook, when Spencer gets his hug from Nug Nug.
In iBreak A World Record when he let's them share in his World Record after ruining their own attempt at it.
Gibby coming to the rescue of the trio during iPsycho is an extremely badass moment of heart-warming.
Anytime we get to see any interaction between Gibby & Guppy, his little brother.
Sam and her mother, Pam, making up at the end of iSam's Mom
"I guess I coulda had a worse kid." "(tearing up) You mean that?"
The ending of iGoodbye, in which Carly watches a montage of iCarly webisodes from the series' beginning to its end as she is in her airplane.
The moment in which a tearful Carly laments not having anyone to go to the father-daughter dance, and then her father walks in through the door, completely dressed in uniform, and he says he will take Carly to the dance. Cue her cheerful response, the studio audience applauding, and the viewers at home weeping Tears of Joy.