This is an interesting question. In the Aristotlean sense of the word tragedy, Winston Smith is not a tragic hero.
First, 1984 is not a tragic drama, which is what Aristotle has in mind.
Second, Aristotle's ideal tragic hero is a person of noble birth (royals like Oedipus or Thyestes) who goes from a position of good fortune to a position of bad fortune. Winston Smith is not a person of noble birth, but rather an "everyman."
Also, Smith does not strike me as someone who is ever in a position of good fortune in this novel. He starts off as mid-level member of the Party and he ends the novel as a mid-level member of the Party and his spirit is completely broken, but he does have, in some ways, a better job than he had previously.
Additionally, Aristotle's "best" tragedy involves the destruction of one family member by another family member (compare son Orestes killing mother Clytemnestra). Winston Smith and Julia are not related and, even though they betray one another, at the end of Orwell's novel they both remain alive.
Thus, I would say that Winston Smith is not a tragic hero in the Aristotlean sense of word. In the modern American sense of the word, however, that would be a different story.
Great answer, thanks a lot!