By our modern standards, you could certainly argue that Othello deserves to die. In fact, you could argue that he deserves worse. At least he got to choose his own death and die with Desdemona near him. You could argue that he should have been condemned and executed without the honor of committing suicide.
After all, what Othello does is reprehensible (to us today). He kills his wife because he thinks she's cheated on him. I think you can easily argue that he gets what he deserves.
Desdemona, on the other hand... I agree with both of the previous posts, especially Post 3.
Generally speaking, I certainly don't think that either character deserves to meet such a violent death at the end of Othello. While some literary critics question Desdemona's actions and/or claim that she didn't do enough to help her own cause or prove her innocence, I still don't think that her actions justify Othello's murder of her. (Even if she HAD been unfaithful to him, would she deserve to die for it?)
I think the bigger issue centers around Othello's suicide. If we consider Othello as a tragic hero (by Aristotle's standards), we know that he must take responsibility for his actions in some form or another. Though I'm not sure we can say he "deserves" to die at his own hands for what he's done, I certainly feel that he himself thinks that suicide is an appropriate punishment for his actions. In his final speech, Othello asks to be remembered as a man who "loved not wisely, but too well" and acknowledges that he "threw a pearl away/ Richer than all his tribe." In these lines, Othello takes responsibility for his actions, acknowledges his mistake, and shortly thereafter, he takes his own life.
Again, considering Aristotle's characteristics of a tragic hero, it seems that Othello felt that he deserved to die for what he did to Desdemona (and Cassio).
Neither character deserves such a tragic ending--that's why it's tragic. Admittedly, Desdemona could have advocated more strongly for herself and simply confessed to Othello that she had lost the handkerchief when it happened. She could have also been more observant of Iago's increasingly suspicious behavior and more cautious when speaking to and about Cassio so eagerly.
Similarly, Othello, a military leader savvy in strategy, should have been able to recognize Cassio's loyalty and Iago's deceit, but neither of the characters' lapses in judgment warrant their violent deaths. Imagine if we all suffered such consequences for doubting a friend or trusting someone too much!
No. neither of them deserved to be killed. Desdemona the least, as she did not do any harm except be faithfult to her husband, and marry him secretly behind her father's back. Othello should not have killed her, but Iago wasso intelligent, that he failed to see if Desdemona was really cheating on him or not. Ohtello too did not diserve to get killed.