What example of young love doesn't contain many examples of hyperbole?!?  Of course, William Shakespeare's play is the perfect example.  It is perfection in the devotion of young love and, therefore, the same perfection in its hyperbole. Hyperbole is one of our figures of speech that involves exaggeration.  In fact, it is often overt, direct, and extreme exaggeration.  It is a "figure" of speech because it is truly meant to be taken "figuratively" and not "literally."  Just keep in mind that an example of hyperbole can also be an example of other types of figurative language as well.  For example, hyperbole can be found within an example of metaphor or simile.  However, for the sake of your question, we will stick only to the figure of speech you need.

We can find examples all over the play of this literary device, but all one has to do is go to the very moment that Romeo notices Juliet for the very best examples.  Then, one simply has to fast forward to the part where Juliet is swooning for the young Romeo to find even more.  The examples above are just a few, but the scenes in question are absolutely overflowing with the hyperbole you mention!  Let's look at a few more.

I couldn't help laughing when going back into these all too familiar lines yet again.  When two people don't really know each other, have only "seen" each other, and are in "love at first sight," they have only one thing to go on:  looks.  So note how every single one of these examples of hyperbole are about the appearance of the beloved.

Juliet's General Appearance:

Arise, fair sun, and kill the envious moon,
Who is already sick and pale with grief
That thou her maid art far more fair than she.

This follows one of your first examples with Romeo naming Juliet as the sun.  Here he is asking his "sun" to "kill" the jealous moon saying that Juliet is "far more fair" than even the moon's beauty.  Hyperbole all found within a metaphor, or an unusual comparison.

Juliet's Eyes:

Two of the fairest stars in all the heaven,
Having some business, do entreat her eyes
To twinkle in their spheres till they return.

Again in the same scene and the very same monologue, Romeo speaks of Juliet's eyes.  Can there be more exaggeration than saying (from afar, mind you) that her eyes are "the fairest stars in all heaven."  These "stars" have left the heaven to settle in his beloved.  Infatuation at its finest.  Again, hyperbole found within yet another metaphor.

Juliet's Cheeks:

What if her eyes were there, they in her head?
The brightness of her cheek would shame those stars
As daylight doth a lamp; her eyes in heaven
Would through the airy region stream so bright
That birds would sing and think it were not night.

So, on to the next part of the body. Romeo says, directly after the star comment, if those starts were taken out of her head and put back in the sky (no longer a part of his beloved), then suddenly "the brightness of her cheek would shame those stars." Due to the brightness of both her starry eyes and her bright cheek, the birds would think it was day instead of night.  Again, complete exaggeration at its finest.  This time it's hyperbole found within a simile, a comparison using like or as.

Juliet's General Appearance (again):

O, speak again, bright angel! for thou art
As glorious to this night, being o'er my head,
As is a winged messenger of heaven

Inspired this time by Juliet's simple sighing, now compares his beloved to an angel who is "glorious" and a messenger with wings "of heaven."  Again, hyperbole.  And in full regalia, here we have hyperbole within both a metaphor and a simile!

Juliet's Denial of Her Family Name:

be but sworn my love,
And I'll no longer be a Capulet.

Here we have Juliet's first example of hyperbole dealing with her beloved.  Noting the blood feud between the Montagues and the Capulets, this is very significant.  Juliet is willing to deny her Capulet blood to become a Montague.  The reader must realize that Juliet will always, always have Capulet blood.  There is no escape from DNA.  Therefore, this is hyperbole.

Romeo's General Appearance:

Retain that dear perfection which he owes
Without that title.

Juliet admits that without the title of "Montague," Romeo would have "that dear perfection."  Not just have it, though, ... "retain it."  This means he had that "dear perfection" even with the dreaded title of her enemy.  Again, ... "perfection" of the beloved.  True hyperbole.

At this point, I have simply given examples within only forty lines of Act II, Scene ii.  This is the very famous "balcony scene."  And I have only focused on the examples of hyperbole BEFORE the lovers notice each other.  this just goes to show just how much young love and, in fact, young infatuation inspires exaggeration.