Two kinds of irony are called into play with the handkerchief. The first is dramatic irony, in which the reader knows more than the characters. We know, for instance, that Emilia stole the handkerchief when Desdemona dropped it and gave it to her husband Iago. We also know through Iago's soliloquies that he will plant the handkerchief in Cassio's quarters. Iago uses this tiny trifle as his main evidence that Desdemona is cheating on Othello:
Trifles light as air
Are to the jealous confirmations strong
As proofs of Holy Writ.
This handkerchief provides the "ocular" proof that Othello demands of Iago, and all the while the reader has followed the true course of the handkerchief from Desdemona to Emilia to Iago to Cassio and finally to Bianca.
Situational irony is also employed here. Othello believes that Desdemona gave the handkerchief to Cassio when he actually is the one who pushes the handkerchief out of her hands as she is trying to soothe him. He is the one truly responsible for the handkerchief winding up in other characters' hands.
Further irony is shown when Othello states that
There's magic in the web of it.
In Act 1, Othello declares to the Senate that the only magic he used to win Desdemona's affections was love. Here in Act 3, though, Othello does a reversal and seems to believe that the love between him and Desdemona is held together by this trifle of a handkerchief, which supposedly had magical powers. This is another example of situational irony in which the cool, rational Othello becomes superstitious and accusatory.