In Othello, why is it ironic that Bianca uses the handkerchief to accuse Cassio of fooling around?
It is ironic because her entrance into a scene that Iago has set up for Othello to overhear him discuss with Cassio his "mistress" (meaning Desdemona) is the icing on the cake that sells Othello on the idea that Cassio is having an affair with is wife.
In this scene, Act IV, scene i, Bianca comes in on this by accident, much to Iago's ironic luck, and says to Cassio:
..what did you mean by that same handerchief you gave me even now? I was a fine fool to take it. . . .This is some minx's token. . . .There! Give it the hobby-horse, wheresoever you had it. . .
And so, Bianca seems to confirm Othello's suspicions about Cassio and Desdemona by alluding to the "minx" that Cassio supposedly got the handkerchief from.
And for some added dramatic irony, the audience is fully aware that Iago gave the handkerchief to Cassio, not Desdemona.
Bianca's accusation is also ironic because she is a courtesan or escort which implies that her living is made from spending time with men who are "fooling around." Because Cassio fools around with her, one would think that Bianca would expect no faithfulness from Cassio, yet she is clearly and ironically jealous when she returns the handkerchief to him.
Similarly, the fact that Bianca, someone who would not have been highly respected in high social classes, has Desdemona's handkerchief is another element of what infuriates Othello, and Iago's plan works even better because Bianca just happens to appear with the handkerchief when Othello is watching. Both examples represent situational irony.