As other’s have suggested, one important purpose of puns in Shakespeare's Romeo and Juliet is to lighten the mood so that the audience remains entertained throughout the play despite the tragic ending they know is coming.

Another important purpose of the puns is to have the audience see the skillful way in which Shakespeare plays with words. He shows off both his ability to craft an interesting story and his creativity with language and words. This also keeps the audience's interest as people listen for the different word games included in the dialog.

Shakespeare often used homonyms, or words that have the same sound but different meanings, to play with words. For example, Mercutio wants Romeo to dance at the Capulet’s party. However, Romeo’s heart is heavy because his beloved (not Juliet yet, but Rosaline) does not return his love. He declines to dance, saying:

"You have dancing shoes
With nimble soles. I have a soul of lead
So stakes me to the ground I cannot move."

Romeo is depressed because of his unrequited love for Rosaline. Shakespeare makes use of the similarity of “sole” and “soul” to explain why Romeo cannot dance. These kind of wordplays held audience member’s attention and also impressed Shakespeare’s chief patron, Queen Elizabeth, who was extremely well-educated and also a bit bawdy herself.

It is also important to bear in mind that audiences for Shakespeare’s plays in Elizabethan England did not sit quietly and listen to the words as modern audiences do. They often participated, applauding as the performance was going on when they liked the events or throwing things at the performers when they did not.

According to A Brief History of the Audience,

“Elizabethan audiences did not know what it meant to be quiet for a performance and would talk back to the actors.”

Thus, incorporating word play in the dialog is another device to keep the audience focused on the words, keep them amused, and elicit their attention to catch all the plays on words.

Tybalt accuses Mercutio of consorting—or associating—with Romeo. Shakespeare uses another play on homonyms that the Elizabethan audience would have understood. Consort could refer to Mercutio’s friendship with Romeo and it could also refer to a consort (group) of musical elements, as in the consort music that was popular at the time. Mercutio’s response to Tybalt illustrates the double entendre.

“Consort? What, does thou make us minstrels? As thou make minstrels of us, Look to hear nothing but discords.”

Shakespeare is showing the two possible uses of the word “consort” and also incorporating the word “cord” as “discord” to play on the theme of music. Mercutio is saying if you make minstrels of us, you will only hear poorly played cords and you will get discord—or a fight or disagreement—back.

There are puns a-plenty in Romeo and Juliet. And thank goodness there are, because otherwise the tone of the play would be unremittingly grim. Because from the opening Chorus we know that Romeo and Juliet, the "star-cross'd lovers", will die a tragic death.

Before we get to that stage, quite a lot happens. Many other characters are involved in the unfolding action and make their mark. Inevitably, some of these characters are going to bring us light relief to take our minds off the terrible fate that awaits the eponymous characters.

One such character is Mercutio, who loves nothing more than to engage in a spot of witty banter with his best friend Romeo. During their regular bouts of badinage puns are liberally employed, as in the following exchange from act 2, scene 4:

Mercutio: Nay, I am the very pink of courtesy.
Romeo: Pink for flower.
Mercutio: Right.
Romeo: Why, then is my pump well flowered. (II, iv, 26-29).

Of course, Romeo and Mercutio, being two highly immature young men, are not talking about flowers, but sex.

Mercutio is such an inveterate punster that he can't even help himself when he's on the brink of death. Fatally wounded by Tybalt in a duel, he still manages to squeeze out one last pun before he leaves this mortal coil:

...ask for me to-morrow, and you shall find me a grave man. (III, i, 94-95).

In this context, "grave" means both "serious", as well as the grave where Mercutio's body will be lying. Though Mercutio's punning doesn't exactly lighten the mood in this tragic scene, it does at least divert our attention from the tragic fate that lies in store for the star-cross'd lovers.

Puns in Shakespeare's tragedies, such as Romeo and Juliet, serve the purpose of lightening the dark mood with humor to keep his audience drawn in and engaged.

In Romeo and Juliet, a few of the characters act as comic relief by making puns.

In the opening scene, Capulet's servants make several puns in their dialogue. Sampson uses the phrase "carry coals" to refer to humiliation. The act of carrying coals was the task that the chambermaid performed, the least significant household servant. Hence, carrying coals was seen as an insulting and demeaning task (Romeo and Juliet, eNotes). Therefore, when Sampson says to Gregory, "On my word, we'll not carry coals," he means to say that he will not allow themselves to be humiliated by the Montagues, making a pun out of the word "coals," which is referring both to literal coals and to humiliation. Gregory twists Sampson's words into a further pun in his reply, "No, for then we should be colliers." The term colliers, pronounced "coalers," refers not to chambermaids, but to "coal miners," forming another pun out of the word "coals." (eNotes).

The character Mercutio is also well known to make puns. In Act I, Scene 4, when Romeo says he feels uneasy about crashing the Capulet's feast due to a dream he had, Mercutio states, "That dreamers often lie," creating a pun with the word lie. Mercutio is using "lie" to refer both to sleepers lying down and to untruths. Hence, Romeo responds with, "In bed asleep, while they do dream things true," meaning, they lie down while being in "bed asleep," and they dream about things that are "true."

Earlier in this same scene, Romeo makes a pun of his own when he says,

Give me a torch: I am not for this ambling;
Being but heavy, I will bear the light.

Romeo is making a pun out of the word light, using it to refer both to literal light from the flame of a "torch" and to lightness of weight. Romeo is saying that since he feels sorrowful, or heavy hearted, he will carry the torch, thereby carrying the light, and becoming lighter in emotion.