For me, the most important thing to note about the use of the handkerchief is that it is a stage prop in the play, and, as such, can be manipulated by the actors in any number of ways. This fact implies the lack of control over the prop's use that any playwright, even Shakespeare, can have, a circumstance quite different from the use of an important object in a work of literature written to be read only, like a story or novel.
In the novel, the object's appearance, use, and subsequent disappearance will be described by the author in detail, according to the significance of the object, and this fact will be unchangeable. However, in a play, the playwright can indicate the presence of a prop and even suggest its importance to the events of the plot, but it will, ultimately, be the use of the prop by the actors that gives the audience the clearest sense of its significance. In this way, the prop's use, even a significant prop like the handkerchief in Othello, cannot ever be fully determined by the playwright.
Let's look at a few of the scenes in which the handkerchief appears to examine how differing uses of it by the characters in the scene could alter the sense of its meaning.
First, it is produced in Act III, scene iii by Desdemona, intended to wipe Othello's face. It falls to the ground (We know this from Othello's text after it drops, line 292: "Let it alone."), and Emilia retrieves it. In the subsequent scene with Iago, she begins with the handkerchief in her possession, and he ends the scene with it in his. But, it is entirely up to the actors (and their director) to decide whether Emilia gives him the handkerchief or he simply takes it from her. Emilia's participation in the deception surrounding the handkerchief is strengthened or weakened, depending upon how the exchange of the handkerchief in this scene is played.
Then, in Act IV, scene i, when Bianca returns to fling the handkerchief at Cassio, it is entirely stage-able to consider that Cassio runs off after Bianca, leaving the handkerchief onstage to be picked up by Iago, who then uses its presence to torment and further enrage Othello. Alternately, Cassio could take the handkerchief off when he pursues Bianca, leaving Iago without this significant ammunition present onstage.
These two scenes are but examples of how the use of any prop, but especially a very significant prop like the handkerchief in Othello, is determined by the choices made in performance by the actors rather than by the playwright.
For more on the handkerchief in these scenes, please follow the links below.