Heroic responsibility is the responsibility to get the job done no matter what. It is described in Methods_Of_Rationality_(fanfiction):
The boy didn't blink. "You could call it heroic responsibility, maybe," Harry Potter said. "Not like the usual sort. It means that whatever happens, no matter what, it's always your fault. Even if you tell Professor McGonagall, she's not responsible for what happens, you are. Following the school rules isn't an excuse, someone else being in charge isn't an excuse, even trying your best isn't an excuse. There just aren't any excuses, you've got to get the job done no matter what." Harry's face tightened. "That's why I say you're not thinking responsibly, Hermione. Thinking that your job is done when you tell Professor McGonagall - that isn't heroine thinking. Like Hannah being beat up is okay then, because it isn't your fault anymore. Being a heroine means your job isn't finished until you've done whatever it takes to protect the other girls, permanently."
- Harry Potter and the Methods of Rationality: Chapter 75 - Self Actualization Final, Responsibility
- The discussion at this Reddit post is excellent.