What are some examples of assonance in Act 3, Scene 5 of Romeo and Juliet?

Assonance is the repetition of vowel sounds and often provides a more melodious sound to a character's speech.  When Juliet describes the nightingale, she says, "Nightly she sings on yon pomegranate tree. / Believe me, love, it was the nightingale" (3.5.4-5).  Here, assonance occurs with the long "e" sound in Nightly, she, tree, Believe, and me.  It makes sense that Juliet's speech would be so musical in this moment because she is happy, but also because she, in part, describes the sound of the nightingale, whose voice is melodious compared to the lark's.  

Further, a few lines later, she says, "Yon light is not daylight, I know it, I" (3.5.12).  Here, assonance occurs with the long "i" sounds in light, daylight, I, and I.  Next, she says that the light "is some meteor that the sun exhales / To be to thee this night a torchbearer, / And light thee on thy way to Mantua. / Therefore stay yet. Thou need’st not to be gone" (3.5.13-16).  Here, the assonance once again occurs with the long "e" sound in meteor, be, thee, thee, and need'st.  She would be pleased by those things associated with night -- nightingales and meteors -- because she knows that Romeo will have to leave when daylight comes.  Therefore, when she describes the night, its sounds and its sights, she does so in quite a musical and euphonious way.  When she recognizes that it is, in fact, daylight, her speech loses much of its musicality.

Can you provide an example of assonance in the prologue to Romeo and Juliet?

As the other educator states, assonance is defined as the repetition of internal vowel sounds in a series of words or phrases.

In the prologue of Shakespeare’s Romeo and Juliet, assonance is used to emphasize the importance of the information to the audience.

Two examples of this come near the end of the prologue:

Is now the two hours’ traffic of our stage;
The which if you with patient ears attend . . . (lines 12–13)

The words “now,” “hours,” and “our” all contain the same sound. This creates a sense of unity within the line itself. In line 13, “which” and “with” both contain the short “i” sound.

Prologues are a feature found in ancient Greek tragedies, which Shakespeare would have known. Understanding that the lines of these ancient Greek plays would have been memorized, then performed, the use of sound devices like assonance help make these lines more memorable. Shakespeare’s use of rhyme and other sound devices like assonance certainly would have helped his Elizabethan actors perform.

Assonance is when you have similar vowel sounds occurring in a number of words.  So it is sort of like rhyming, except it does not generally come at the end of a word.

I think you can see examples of assonance right at the start of the Prologue.  For example, Shakespeare writes

Two households, both alike in dignity,

To me, the "o" sound in "holds" is the same as the one in "both."  I also think that the "i" sound in "in" is the same as the second one in "dignity."  So, to me, both of those are examples of assonance.