The first one I would consider, which is simply 4-5 lines each back and forth would be the last 20 lines or so of Act I, scene 2.  It's between Romeo and Benvolio and they are discussing crashing the Capulet party where Romeo can compare Rosaline's beauty to others at the party.

The next is Act II, scene 2, where Juliet starts with the famous "Romeo, Romeo, wherefore art thou Romeo" line. They go back and forth through the famous balcony scene, and it would be fun to do as well.

Another scene that would be fun would be the bantering back and forth between Mercutio and Benvolio in the very beginning of Act II, scene 4.  Neither one knows that Romeo is now in love with Juliet, and both are worried about Tybalt's challenge to Romeo.  This would be good since they go quickly back and forth as well--and Mercutio is always a fun character to portray. 

Hope that helps!  Good luck!

I linked below to 2 scenes where Romeo and Juliet have extended back and forth dialogue. The first scene is the famous balcony scene where Romeo has come to see Juliet at night. The second occurs during the party when Romeo and Juliet first meet. They are certainly both over 20 Shakespearean lines back and forth, but I'm not sure if you mean lines of Shakespeare or lines in the acting sense of the word.

If it is using the acting term "lines" and not referring to lines of text, I'm not sure if the pieces will be long enough for your project or if you will be able to find that many lines between just two characters in the text.