The handkerchief in Othello is one of the most famous props in all of Shakespeare. So, it's comings and goings, tracked by way of stage directions, are very important to the plot of the play.
In Act III, scene iii, my Arden edition of the text gives the stage direction as "She drops her handkerchief," after Othello's line "Your napkin is too little." Almost immediately, Desdemona and Othello exit, and Emilia, alone onstage, picks it up. She says:
I am glad I have found this napkin. . .
My wayward husband hath a hundred times
Woo'd me to steal it. . .
Iago enters and manages to take it, apparently against Emilia wishes, who says:
Give me 't again, poor lady, she'll run mad
When she shall lack it.
But the deed is done. The handkerchief is in the hands of Iago and will become a key prop in his further insistence to Othello that his wife is being unfaithful with Cassio.