In Acts I and V of Othello, Shakespeare gives Othello dignity and heroic words, but in Acts II, III, and IV he gives Othello ignoble and immoral actions. Overall, Othello wins a battle with words but loses the war overall. As such, he is no hero--only a tragic one.
In Act I, Othello defeats Brabantio's, Iago's, and Roderigo's plans to annul his marriage, place him in prison, and do him physical harm. How does he win? Using his words, and not using his hands. On the street, he talks Brabantio's men--who have him outnumbered--out of fighting. He says:
Later, in the Senate with the Duke, he likewise wins Desdemona's hand by using powerful rhetoric:
When I did speak of some distressful stroke
That my youth suffer'd. My story being done,
She gave me for my pains a world of sighs:
She swore, in faith, twas strange, 'twas passing strange,
'Twas pitiful, 'twas wondrous pitiful:
Just as Othello used stories to win over Brabantio and Desdemona, so too does he use a story to win over the Duke, the Senate, and us. The marriage is condoned, Othello is commissioned to Cyprus, and Desdemona may accompany her husband on the honeymoon. At this point in the play, Othello looks like a hero. But wait...
In Acts II-IV, Othello loses his power of language. He is like Sampson with his hair cut off: powerless. He suffers seizures, rage, fits of jealousy, misogyny. He slaps Desdemona publicly, slanders her name, murders her, aligns himself with a villain, and shows little remorse for his crimes. Listen to the once mighty Othello now:
He's a stuttering, powerless fool. Iago-the spider-has Othello-the fly-trapped in his web of deceit and jealousy. Iago is a puppet-master and Othello now dangles on his strings. Othello is a green-eyed monster. A beast. A slave.
In Act V, Othello tries to resurrect his reputation, but it's all for show. He knows that he is ignoble, immoral, a misogynist, and a murderer. His last monologue about doing the state "some service" is all lip service. He still only focuses on himself. He makes no confession or prayer regarding the two women who lie dead on the bed. He does not honor them, only himself.
No, Othello suffers too much pride to become heroic. He only seems heroic with his words.
At the outset of the play, Othello describes the way that Desdemona fell in love with him and he gives some insight into a background that could be called heroic. He was sold into slavery when he was young, traveled the world and saw all kinds of horrifying things, and became a general of great renown through his feats on the battlefield. He demonstrates as well a type of humility, even as he expounds upon his feats that help to build a sense of dignity as well. He appears to be a confident, capable and powerful general at the outset of the story.
Of course we see the devolution he goes through as Iago uses and abuses him, but in his final speech, he does perhaps regain some of that dignity.
He appeals to the fact that he has "done the state some service" so rather than arrest him they let him kill himself. He also asks them to tell his story as one who found something precious but threw it away, not realizing what it was. Once his madness has passed, he does regain some dignity, but certainly not a full measure of his previous stature.