Compared to the modern world today, to what extent is this play realistic, and to what extent is it unrealistic?
Naturally, there are going to be aspects of such a profound work that have relevance and some elements that could experience a more modern sense of "troubleshooting." The notion of intense jealously, scheming, and manipulation are all relevant to modern times. Iago's feeling of resentment at being passed over and his desire to impact everyone around him so that they experience a similar level of misery is something that is relevant today. Adolescents experience this constantly as do adults. This element is quite valid. Another aspect which is powerfully relevant is Othello's misapplication of love fused with doubt and personal insecurity. In many instances today, the same sense of fear in relationships does infect the bond of love, corrupting it to a point where there might be little left. This is definitely valid. In terms of what might be modified by today's standards would be that there are other intervening factors that would limit Iago's destruction. In the modern setting, there is trust and reliance on counselors but with the proliferation of information from a variety of sources, the sheer dependence that Iago enjoys from Othello might be something that could stand some diluting in the modern setting. I also think that Desdemona's status in the deteriorating relationship with Othello might be altered by modern conditions, as well. Whereas silence and misunderstanding seem to dominate in the drama, perhaps the modern notion of dialogue and a greater sense of emotional transparency would be present in the modern predicament. This is not to dismiss what happened between them as something that could not happen today, but rather explores different avenues that could have been taken to prevent it to the extent shown in the play. This is not "unrealistic," as much as explored in a different way in the modern condition.
Realistic aspects: Shakespeare uses Othello's character to demonstrate how the elite in society will often dine and socialize with those who are normally outcasts because it makes them look more charitable and urbane, but when it comes to being true friends with those who are different, the elite draw the line. Desdemona's father Brabantio represents this hypocrisy; he regularly invites Othello to dinner and enjoys his company, but he would rather his daughter marry a dunce (Roderigo) than marry Othello (a black man).
While Iago is an extreme version of a master manipulator, humans like him still exist. They are those people who are willing to go to any length to accomplish their goals. They study the weaknesses of others and then take advantage of those weaknesses whenever it suits them to do so.
Unrealistic elements: Desdemona's character is rather unrealistic, especially for modern society. Most women in her position would defend themselves and be observant enough to realize what they could do to correct the situation. While some women do stay in abusive relationships because they feel there is no escape, a modern Desdemona would have many more options for defending her character and saving herself than she would have had during Othello's day.
Othello's complete gullibility is also an exaggeration. It is difficult to believe that someone of his military reputation and accomplishment would not be able to see the truth about one of his own men (Iago). Othello built his reputation on being a fierce fighter and effective strategist, qualities which just seem to disappear when Iago is around. Granted, relationships with women do seem to make men do crazy things :), but Iago's complete control over someone who is so experienced in reading others is extreme.