Heroic Lunatics are Heroes with Mental Illness or Mental Disorder are heroes whom suffers from a disorder that mainly affect the personality and behavior of the Hero. A hero with mental illness does not mean that the hero is insane. The most popular mental disorders are amnesia and depression. Examples can be;

•  Max Payne who suffers from depression.
•  Rorschach who has a Paranoid Personality Disorder. 

These characters are also Tragic, since the trait of being mentally unstable is being unable to tell right from wrong. Examples include Ryūko Matoi, Homer Simpson, Shoto Todoroki, Goku, Sinon and Joe Swanson.

Nevertheless, some can also be Pure Good if they have no corrupting factors and if their mental illness doesn't interfere with their moral agency and knowledge of right and wrong. Maria Robotnik and Rayman are good examples for this.

Notes

• Not all heroes are mentally unstable when they first appear. While some are, others who were sane to begin with suffer experiences that causes their sanity to snap. The primary causes for this are prolonged abuse and humiliation. A severe-enough degree of fear can do it too, but less commonly.
• Mentally ill heroes can only be Pure Good if they make it clear that they have an understanding of what is right and wrong and meet all the criteria, even in spite of their limited views on reality.
• Non-humanoid entities and angels never count, regardless of how good they can be. To them, such acts are normal or a simple spiritual/cultural function.

See Also

Mentally Ill on the Villains Wiki.