Elizabethan audiences would laugh at the constant references to Jessica as a boy. She has decided to elope with Lorenzo, and to hide her identity, she will be disguised as a boy.
The humour does not only lie in the fact that the characters make repeated references to Jessica being a boy, which in itself is funny, but also because of the restrictions applicable to actors during Elizabethan times. Women were not allowed on stage and thus female roles had to be played by young men or boys who would be appropriately dressed and made up to look like girls or women.
The audience would therefore roar with laughter each time there is talk of Jessica being a boy, for she would in fact, be a boy! In this instance she would be a boy playing a girl dressed up as a boy.
The double irony and the wordplay (pun) would not be lost to the audience and would generate great mirth.
With this in mind, you can imagine being a member of the audience and hearing the following repartee:
Jessica says: "I am glad 'tis night, you do not look on me,
For I am much ashamed of my exchange:"
And: "Cupid himself would blush
To see me thus transformed to a boy."
"What, must I hold a candle to my shames?
They in themselves, good-sooth, are too light.
Why, 'tis an office of discovery, love;
And I should be obscured."
Lorenzo replies: "So are you, sweet,
Even in the lovely garnish of a boy."
Where would he double irony come into exactly?
The first irony lies in the fact that a boy plays the role of a girl.
The second irony is that the "girl" is disguised as a boy when she is in actual fact, a boy!