Heroic responsibility is the responsibility to get the job done no matter what, including not shifting any responsibility for its completion on to others.
“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.” In Harry’s voice was a touch of the steel he had acquired since the day Fawkes had been on his shoulder. “You can’t think as if just following the rules means you’ve done your duty. –HPMOR, chapter 75.
The discussion at this Reddit post is excellent. This wiki requires work.