Doubles as Tear Jerker. Anton Vanko is dying at the beginning of the movie, watching Tony on the TV. He heartbreakingly tells Ivan "That should have been you", and apologizes to Ivan for not giving him a better life, but giving him the knowledge he could use to achieve it. Despite what must have been a really crappy life, Anton loved his son, and Ivan loved him back just as much.
Also, Ivan tells his father to pay no mind to that crap. He likely didn't care about wealth or power, but what he DID mind was his father dying of alcoholism and depression and in poverty because of what Howard did to him.
Tony complains about the Parental Abandonment issues he had with his father, which makes Howard Stark's recording admitting that Tony was his greatest creation into such a powerful moment. That and the outtakes of the recording, which acts as a posthumous Like Father, Like Son moment as we see the elder Stark was a lot like Tony.
The reveal in Captain America: The Winter Soldier paints Howard's actions in a completely different light. The day Tony went to private school was his happiest not because he got rid of a kid he didn't care for. He knew HYDRA had infiltrated SHIELD and was trying to kill him so he moved his son out of harm's way.
This one takes a bit of Fridge Brilliance. After Rhodey steals the Mark II Iron Man suit, Natasha mentions that Tony's security system is designed to prevent unauthorized use of the suits. Thought it's never said explicitly, it's implied that Tony had been planning for Rhodes to inherit the Iron Man title and machinery after he died.
Tony is running flying for his life from his best friend in another suit of armor that has been commandeered by the bad guy. The Expo is being evacuated because of the drones attacking. One brave little boy in an Iron Man mask and repulsor toy puts his hand up to fire at the drone. The drone mistakes the kid for Tony and takes aim as well. Then Tony lands, takes out the drone with his repulsor, then looks at the little boy, says "Nice work, kid," and goes back to being chased.
Making this absolutely Heartwarming in Hindsight is that the kid was none other than a young Peter Parker before he became Spider-Man.
Borrowed from the WMG page: despite everything that happened between them afterwards, it is still fairly moving to realize that Howard Stark may well have named his son 'Anthony' after his former partner, Anton Vanko.
Turns out the drones have been set to explode after the final battle. Tony himself is surrounded, but what is the first thing to come out of his mouth?
Tony's pitiful attempt at making an in-flight meal for Pepper after the Monaco attack. As he put it later, he'd been planning to make her an omelet and tell her that he was dying. Like with the strawberries, the omelet was a clumsy, failed attempt to make her feel better. It took him three hours to make, and he never even really got any credit for trying.
The ending where the senator has to give Stark and Rhodes medals is mostly just funny, but when he thanks Rhodes and tells him he deserves the award he's being totally sincere, despite the fact that Rhodes had done his best to publicly undermine him earlier in the film.
The first time we see a sincere side of Natasha, when she listens to Tony lamenting about his eventual demise. She even gives him an honest answer to his question. The moment gets slightly funny after he takes it literally.
Tony: Can I ask you something personal? If this was the last birthday party you were going to have, what would you do?
Natalie Rushman / Natasha Romanoff / Black Widow: I would do whatever I wanted to do, with whomever I wanted to do it with.
Just the amount of screentime we get of Natasha and Pepper bonding; such a shame we never got to see any more of that beyond this film.