Tony Stark upgrades the arc reactor keeping his heart beating and instructs Pepper Potts to junk the old one, stating that he's not a sentimental man. However, instead of throwing it away, Potts turns the old reactor into a trophy with the words "Proof that Tony Stark has a heart." inscribed in it.
Also extremely lucky for Tony that Pepper decided to save the old reactor, as if she hadn't, then Tony would have died because Stane stole his current one.
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.
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.
"You're all I have too, you know."
The friendship that develops between Yinsen and Tony in spite of the fact that it ends with a Tear Jerker.
Tony (and in only his second battle as Iron Man, no less) saving a group of refugees from the Ten Rings, particularly arriving just in time to save a father from being killed in front of his family.
Doubly so when the little boy runs up to hug his father. A small moment in the grand-scheme of the overall ass-kicking, but definitely heartwarming.
Heck, Tony's entire development in the beginning half of the movie, showing us that despite being an irresponsible, womanizing Insufferable Genius, he hates to see people make sacrifices for his own sake and wants to do all he can to prevent any needless fatalities he may indirectly cause. It rings all the more true after what happened to Yinsen and the convoy at the film's start.
Tony's Tranquil Fury while watching the people from Yinsen's hometown suffer and die on CNN, culminated by him shattering a wall in silent-rage, shows us truly for the first time that there beats a compassionate heart behind all that ice-cold sarcasm.
It's not just a wall: it's a glass wall, and he isn't shattering the glass, he's shattering his reflection.
When Rhodey comes in and finds Tony on the floor of the workshop, the first thing he says after a near-death experience, almost completely out of breath is: "Pepper! Where's Pepper?"
Also earlier in the movie when Stane takes the arc reactor; the utter terror in Tony's eyes when he tells him Pepper is next on his hit list. Really, Pepper and Tony's entire relationship is one giant CMOH crossing over with a frequent CMOF.
Listen close: there are two instances where you can just barely hear Pepper's voice calling Tony's name. The first is when the Ten Rings are dunking Tony's head in a barrel full of water. The second is after the battle with Stane. Tony's arc reactor is barely flickering, and then we hear Pepper's voice.
After falling out of the jeep and into the middle of an intense firefight, the first thing Tony does is to start furiously dialing his phone. Since he's interrupted by the bomb before he can finish, the audience never finds out who he was trying to call? Rhodey or Pepper?
Probably Rhodey, since he was also in Afghanistan, and not far from where Tony was. ...but still...