In the season three finale, Madeline gets one of these when she defends her son to some FBI agents in the face of some pretty damning evidence. That she comes off as the most Bad Ass person in the room helps.
She gets another in the next episode, when Michael, distraught by Simon's predictions that Michael will become He Who Fights Monsters, breaks down while explaining it to her. In spite of her fears about her son's work, Madeline hugs him and reassures him that he's a good man.
Sam in the season one finale, sending a coded message in his hostage proof-of-life photo to Michael and Fiona . . . not to come rescue him. Michael, of course, comes anyway.
When he's reduced to asking his captors to kill him, and they don't, the look on Sam's face is heartbreaking. It wasn't being tied up and tortured for hours that broke him, it was being unable to save his friends while being tied up and tortured. Note that he very nearly succeeded in that attempt.
Maddie coming back from Tampa in "Guilty as Charged", because Michael asked her, and staying to convince Jesse not to shoot Michael.
While perhaps not your normal definition of heartwarming, but Maddie chewing out Jesse and Michael at the same time and speaking to them as family (to her, to each other).
Fiona playing toy soldiers with the little boy who might be dying in "Do No Harm". "You're going to need some tactical support," indeed. D'awwwwww...
And later in the same episode, when a female medical scam artist is laughing about comforting the parents of the children she's allowed to die, Fiona can't stop herself from beating the woman up. So heartwarming in a very Fiona way.
Michael's ex-fiancee shows up and there's a lot of awkward confrontation with Fiona. At the end of the episode, Michael explains why he didn't marry her. He was in Ireland, and he met a fiery lass who complicated his life, and "you don't marry someone when you're in love with someone else".
Sam needs some help with an old buddy in prison who's life is in danger from an extra-crooked gang member. Mike points out Sam can't go in undercover, as he'd already visited the prison, and offers to do it himself.
Jesse and Maddie in the season six premiere. Especially when you realize Jesse knew Maddie was up in the attic and safe and he was simply trying to clear the house. Their playful banter is similar to that of Maddie and Michael and shows how close they are. It also gives the two characters a chance to show how awesome they are.
Michael in Tranquil Fury mode dislocates someone's arm because...
"That was for my mom."
Goes to show how much closer Michael has become towards Maddie and his protectiveness of her. Also takes on a slight double meaning since said person intimidated Maddie prior and so there's a certain subtext that Michael's standing up to a surrogate Frank.
Even though a few minutes later he's revealed to be the bad guy, Card's "two bottles" speech to Maddie is oddly heartwarming. She's blaming herself for everything that happened, and Card helps her to understand that some things are just beyond anyone's control. Card seems to genuinely care about Michael and his family, despite giving the order to kill him.
In CIA custody, facing tremendous jailtime, and treatment as a terrorist, Sam continues to lie to the CIA to try and help Michael.
And then Michael refuses to abandon Sam, and goes to rescue him. And while Fi and Jesse are justifiably concerned about the CIA, this is one thing they all agree on.
A great one for a minor character: the thief Tommy in "Fearless Leader" is being used by Team Westen to try and trap a much bigger criminal. Team Westen pretends to be a crew of small-time thieves who work with Tommy to get close to a drug robbery the bad guy is planning. They eavesdrop on the job's planning which involves Team Westen ending up being the Cannon Fodder to hold off a small army of drug dealers with automatic weapons while the badguys get away scot-free. Then, when Tommy meets with Mike later to discuss the job, he does something that completely shocks Mike: Tommy calls off the job and tells Mike to get his "crew" out of town because they'll get killed, and he cares too much about his new friends to let them get gunned down. Its so shocking that Mike takes a gamble and drops the whole facade, and he and Tommy team up to take the villain down - all because, despite the fact that Tommy's a thief, he's a decent person, which Mike never expected to come across.
In Brothers in Arms, Fi and Maddie kidnap a Russian spy and trick his superiors into thinking he had leaked secrets in order to help Michael infiltrate a secret Russian prison. However, they find out that the spy they just intentionally burned isn't such a bad guy. He became a spy since he had no other choice, and uses the money to help pay to put his girlfriend through nursing school. However, now that his own government thinks he betrayed them, he'll be executed if he is caught. Fi and Maddie instead arrange for the spy to be smuggled out of the country and reunite with his girlfriend, stating that they believe in giving former spies a second chance at life.
Madeline making the ultimate Heroic Sacrifice of the series, leaving Michael free to escape and secure everyone's futures.
The simple fact that Barry attends Michael's fake funeral. There's something poignant and ridiculous about that.
Also, the CIA putting up a memorial star for Michael after his apparent death, which means that they officially recognize him as a member of the CIA again, even if it's only done "posthumously".
Michael and Fiona escaping from Miami. For one of the darkest seasons of the series, it's refreshing to see the two who were at the lowest point of their relationship in a position to start a new life together.
Poached from the Fridge, but it does speak the point better. The series finale gives an in-universe explanation for Michael's narration: He's actually recounting his life story to Charlie
In the series finale, every character re-creates their opening intro. Fi asks "Shall we shoot them?". Sam says "You know spies..." and dives away from an explosion. Jesse pops two guys after an explosion. Maddie tells him that people need his help.
Michael finally getting reinstated to the CIA and his burn notice rescinded, along with him and Fiona being honored with an Intelligence Star and a place on the CIA Memorial Wall where officers that have died in the line of duty are memorialised. After the massive falling out in the series finale, the CIA recognises Michael "died" trying to do his job that he eventually became disenchanted with, and at the end, Michael receives a military funeral.
At all military funerals, the burial party will have an enlisted soldier fire the rifle and stand guard over the grave. For Michael's funeral, the rifleman and guard is Commander Samuel Axe. And once the funeral is over, Sam and Jesse head back to Miami - because there's someone who needs their help, so they're going to carry on their work.