In the second Scarecrow sequence, Batman is forced to remember the death of his parents. While he walks off the memories, he remembers his first meeting with Jim Gordon, then a regular cop. Unlike the jackass who thought little Bruce would be fine with all the money he had, Gordon was sympathetic, and says that Bruce can call him Jim. It adds a layer or two to Batman and Gordon's friendship.
When you first enter the Sanatorium, you see Dr. Cassidy "escorted" by a thug to where the other hostages are being held, where she kneels sobbing and scared. If you go nearby her prior to eliminating all of the thugs, she'll stop sobbing and whisper "Batman?" who will tell her to stay put since it's not safe yet. For the rest of the time you're busy eliminating the thugs, she isn't crying anymore.
Especially heartwarming when you realize that he saved her from Zsasz cutting her into pieces by showing up Just in Time when Zsasz kidnapped her before the game began. He's saved her life before, and she's trusting him to save it now.
The depiction of the actual rescue◊ in the pre-game comic The Road to Arkham definitely qualifies. If that doesn't touch your heart, you must be made of stone.
After you first escape from the Intensive Treatment building, the grounds of the asylum will still be full of guards patrolling. If you take your time to talk to them, you will learn from one of them that you have perviously saved his life a few years ago from Two Face and another will tell you that his son is your biggest fan. It is heartwarming to know that the people you helped remembers your deeds. Also a Tear Jerker, as when you return to the area later on, you will find them all killed by the escaped thugs.