What is the significance of the rhyming couplets at the end of many scenes throughout Othello? Particularly Iago's use of a rhyming couplet at the end of act 2, scene 1.

The rhyming couplet at the end of a scene is found frequently in Shakespeare's works overall, not just in Othello. Usually it has the effect of summarizing the thoughts or the general meaning expressed in a soliloquy or in a scene or act as a whole. In this particular scene of Othello, Iago, in his usual nasty-spirited and voluble way, is talking about his plan to make Othello believe that Desdemona has been unfaithful to him with Cassio. The summation in the last two lines, "'Tis here but yet confused. / Knavery's plain face is never seen till used" is like a compressed paraphrase of his overall point that although one would think his (Iago's) deceptive intentions (the "knavery") would be obvious to an observer, it's only when the plans begin to be executed that other people will figure out what's actually happening. The rhyme gives an added punch to the thought.
Similarly, in Hamlet act 5, scene 1, after Hamlet and Laertes have fought and been separated ("plucked asunder"), Hamlet says,

Let Hercules himself do what he may,
The cat will mew, and dog will have his day.

Hamlet's point, open to interpretation of course, could be that justice—that is, Hamlet's view of what is just—will prevail regardless of whatever deceptions and wrongs the King and others perpetrate. The couplet summarizes not only Hamlet's thoughts in this scene, but his mindset throughout the entire play. Interestingly, because Shakespeare and his contemporaries use predominantly blank verse, the appearance of these couplets has a special, arresting effect. By contrast in, for example, the French tragedies of Racine such as Phèdre, which are all in rhyme, this kind of special effect cannot occur, in spite of the other virtues in Racine's dramas.