More specifically, what effect does the use of hyperbole have in Act 2, Scene 2 of Romeo and Juliet?

The hyperbole in Act 2/ Scene 2, that stands out is most certainly;

"Had I it written, I would tear the word."

The impact of this statement is important because Romeo is relating that he wants Juliet to love him for himself, for who he is...... not for his name to influence her in any way; he wants her to love the person he is.

As far as relating that to Shakepeare, I'm not sure what you mean. Shakepeare's desire for people to love his work for the work itself?