When you think about a conventional (or typical) hero, what qualities come to mind? I think of brave, physically attractive, selfless or in some way sacrificial of himself for others, someone who is naturally goodhearted, talented, a good fighter, often self-made, and a good decision maker. Also, most people associate heroes with "winning" in the end. They fight and defeat the enemy (who is typically a threat to the greater good), win the love of the lady, and end up happily ever after.
In many ways, the conventional hero possess qualities that are the result of both age and experience. Romeo, because of his youth, immediately lacks the wisdom that comes from age and experience.
Additionally, though Romeo is physically attractive (to Juliet at least) and skilled with a sword, he does not come across as self-less nor does he seem to aim at goodness. In fact, he's just the opposite. He is quite spoiled and selfish, and all of his decisions are based on his desire for instant gratification. He is not a planner. He is rash, hasty, emotional, and fickle in decision making. After killing Tybalt, he ends up in the Friar's cell emotional and crying to the point that the Friar calls him "womanish." This is not the courage and fearlessness we expect of a conventional hero.
Finally, Romeo does not win in the end. He does fight his enemy which are only a threat to the Montagues and not to society, but does not necessarily defeat him. He does win the hand of Juliet, but at a very great price. Rather than living happily ever after, in the end both die.