Iron Man/Heartwarming

Comics
"Rhodey, I realized something else that night. In my own way, I'm as helpless and scared and ignorant as the baby. I need help."
 * In Iron Man #182, Tony Stark had been living on the streets drunk. After helping deliver a baby and nearly freezing to death, he lay in his hospital bed telling Rhodey about his decision to quit drinking.

Iron Man

 * Tony Stark upgrades the arc generator keeping his heart beating and instructs Pepper Potts to junk the old generator, stating that he's not a sentimental man. However, instead of throwing it away, Potts turns the old generator into a trophy with the words "Proof that Tony Stark has a heart." inscribed in it.
 * Stark is regularly seen ridiculing a particular AI-controlled robotic assistant, often unsatisfied with its work and at one point threatening to donate it to a city college. However, when a dying Stark tries to reach for the old arc generator Potts saved for him, "Dummy" loyally grabs it for him and hands it over. The following shot of the robot makes it look almost like a concerned face. As Stark reaches for it, he whispers "good boy". Cue the sympathetic "Aaawwwwww" from the crowd.
 * The photos of Tony as a child show that a robotic arm was the first AI he ever built. If the arm itself is "Dummy", (we have no reason to doubt otherwise), then despite how much he belittles it, he still has kept it all these years.
 * Earlier in the film, when Tony is picked up in the desert after his escape, when Rhodes rushes out of the chopper, hugs him, and says: "Next time, you ride with me, okay?"
 * It's really the body language that does it - Rhodes kneels down in front of him and Tony just sort of slumps on his shoulder. It's quite clearly the first time in months that he's let himself relax.
 * Also, when Tony arrives back in the states and reunites with Pepper he notes that she's crying tears of joy. The fact that they both snark about it does nothing to diminish the moment.
 * The moment where Pepper and Tony are laughing in relief after Pepper manages to switch the arc generator from the old one to the new one without causing any serious damage. She asks Tony not to ask her to do that again. Suddenly, and surprisingly earnest, Tony says "But I don't have anyone but you." It could so easily have been sappy, but then Tony pulls an embarrassed "Oh my God, I can't believe I just said that" face which, oddly enough, makes it even more heartwarming. In fact, with this movie it seems that when it's not being jaw-droppingly awesome it's being genuinely heartwarming instead.
 * The friendship that develops between Yinsen and Tony

Iron Man 2
"Tony: Pepper!
 * 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 Tear Jerker. 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.
 * This one takes a bit of Fridge Logic. After Rhodey, Romanov 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 to make Rhodes into the next Iron Man
 * Tony is running flying for his life . 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. Tony lands the suit, takes out the drone with his repulsor, then looks at the little boy, says "Nice work, kid," and goes back to being chased.
 * 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.
 * Turns out the  have been set to explode after the final battle. Tony himself is surrounded, butwhat is the first thing to come out of his mouth?

flies off to save her"