In my mind, one instant connection between the themes of Othello and modern times would be the underlying dynamic of how human beings grasp the realities of insecurity and doubt. Othello, as a character, could not handle the realities of insecurity and did not effectively confront doubt. This is something that can be seen in the modern setting, in that individuals struggle with how to appropriately address issues of doubt and insecurity. What does one do when they suspect the one they love? Othello does not know how to deal with this, and I suspect that more than a handful of people in the modern setting still struggle with this.
Such an idea becomes an extrapolation of the fundamental thematic element of the "insider/ outsider" in the work. Othello is an "outsider" in many respects. He is a soldier. He is a man of color. He is not of established wealth and power. While he is taken in as an "insider," a sign that he has "made it," he, himself, still sees himself as an "outsider." He fails to fully appropriate this sensibility in himself, which is why the insinuations of Desdemona's unfaithfulness strikes at him with so much ferocity. Certainly, Iago understands that while Othello might be an accepted insider, he still sees himself as an outsider and he strikes at this with repeated success. This is something that can be seen in the modern setting today, as there are significant demarcations in different realms between "insider" and "outsider." The stratification used to construct such a reality is not only external, as it is absorbed by the individual in how they view both their own reality and their own consciousness.
In addition to the excellent answer provided above, I would like to point out that villains and villainy are not confined to plays by Shakespeare. There are still plenty of villains around in our modern society, and Shakespeare's Iago, as well as some of his other villains, can show us, and warn us, how they operate. Typically they are friendly, often likable and disarming. Hamlet says of Claudius that "a man may smile and smile and be a villain." What is characteristic of all villains is duplicity. They pretend to be one thing and are really something else. Lady Macbeth tells her husband: "Look like the innocent flower, but be the serpent under it." Shakespeare's Iago should teach us many things. We shouldn't be too quick to make friends. We shouldn't be too trusting. In Shakespeare's Julius Caesar, Caesar tells Antony why he distrusts and fears Cassius and ends by saying of Cassius what is true of all villains: "Such men can be very dangerous."