Consider the opening lines from the first scene of the play Damon and Pithias, written by one Richard Edwards in 1571; the lines are spoken by a Pleasant Gentleman by the name of Aristippus:
Though strange (perhaps) it seems to some
That I, Aristippus, a courtier am become:
A philosopher of late, not of the meanest name,
But now to the courtly behaviour my life I frame.
Muse he that list; to you of good skill
I say that I am a philosopher still.
The entire play is written in rhyming couplets. Listening to Dr. Seuss is fun for 15 minutes, but would be tiring to put up with for three hours; so it is with an entire play written in rhyme (the fact that the lines have no consistent meter just makes things worse).
So, happily, such exercises became rare very quickly.
But Elizabethan dramatists did use rhyming couplets, albeit sparingly. Most often, a rhyming couplet might be used to end a scene; less frequently, the last lines of an important speech, or an individual’s last lines in a scene, might be spoken in a rhyming couplet. The authors correctly sensed that a cute rhyme in iambic pentameter would give a feeling of ritualistic finality to a scene. Often such a rhyme allowed a speaker to conclude his or her thoughts with a pithy sentiment.
The first scene of Julius Caesar does not end with a rhyming couplet; but the second scene does:
And after this let Caesar seat him sure;
For we will shake him, or worse days endure.