In order for Mr. Rochester to be a tragic hero in Aristotelian terms, he must meet three specific criteria: first, he must be a very important man to community and possibly country; second, he must make a tragic mistake or have a tragic flaw that leads him to tragedy; and third, he must suffer a shocking punishment, not necessarily death (i.e., Oedipus), for the tragedy he causes. The Shakespearean tragic hero introduces death as the only right punishment. In one view, then, under these three criteria, Mr. Rochester is not a tragic hero according to Aristotle because he does not meet them (nor the Shakespearean criterion of death).
An Important Man: Mr. Rochester is a very wealthy man but he is not important to any community because although he is social, he is not one going about the country winning battles for war or bringing to himself great glory like Macbeth or Othello did. Mr. Rochester keeps to himself, minds his own business, and travels abroad.
A Tragic Flaw:Mr. Rochester has many flaws, not just one that leads him to death. He lives a secretive life by not telling one person the whole story of his life. He then manipulates others for his own entertainment or control, but this isn't the reason that his house burns to the ground and he is maimed in the end. One might argue that keeping his wife locked up in the attic is his tragic flaw because she breaks out and starts chaos such as the story's ending fire; but that does not fit Arostotle's criteria because the flaw must be a characteristic of personality such as ambition, greed, or paranoia.
Hero is Punished by Tragic Flaw: Mr. Rochester does not die in the end because of any personality flaw; rather, he is shockingly punished by being maimed and blinded, yet he is granted the opportunity to redeem himself and live happily with Jane.
Summarily, Mr. Rochester might be considered more of a Gothic Hero! He is victimized by his father, brother, and his first wife's family by being tricked into a marriage with a lunatic. Then, terrifying and melodramatic scenes unfold throughout his life. These, among other elements of Gothic Literature, are just a few of the reasons why he is not a tragic hero.