What is an example of foreshadowing in Romeo and Juliet?

Shakespeare's Romeo and Juliet doesn't begin with foreshadowing, but with foretelling.

CHORUS. ...From forth the fatal loins of these two foes
A pair of star-cross'd lovers take their life;
Whose misadventur'd piteous overthrows
Doth, with their death, bury their parents’ strife. (Prologue, 5-8)

The Chorus tells the audience what's ultimately going to happen. There's no mystery about that. What the audience doesn't know is how it's going to happen, or if anything else unrelated to the "star-cross'd" lovers' deaths might happen along the way.

However, Shakespeare does manages to infuse a little bit of foreshadowing into the Chorus's foretelling of the end of the play.

CHORUS. A pair of star-cross'd lovers take their life... (Prologue 5)

This is a very subtle play on words, the meaning of which becomes clear only when Romeo and Juliet take their own lives—if anybody actually remembers what the Chorus said in the fifth line of the play while they're watching Romeo and Juliet kill themselves "two hours" later.

With this Prologue, Shakespeare sets up quite a challenge for himself. Since he tells the audience what happens at the end of the story, Shakespeare has to write a really good play to keep the audience's interest, particularly since the basic story of Romeo and Juliet was well-known to his audience.

There had been three fairly recent publications of the story of Romeo and Juliet, including Arthur Brooke’s narrative poem, The Tragicall Historye of Romeus and Juliet, published in 1562—which scholars consider the primary source of Shakespeare's play—as well as Giuletta e Romeo, by Matteo Bandello, published in 1554, and Luigi da Porto’s Giulietta e Romeo, published in 1530. Other theatre companies in London were also performing their own versions of the Romeo and Juliet story.

Nevertheless, the question arises that if the audience knows what's ultimately going to happen, is the foreshadowing in the play really foreshadowing, or is it simply a reminder of what the audience has already been told?

Most of the faux-foreshadowing in the play involves the deaths of Romeo and Juliet, but even what seems like foreshadowing of events affecting other characters really isn't.

In act 3, scene 1, Benvolio cautions Mercutio that that they should get out of the streets to avoid meeting any Capulets:

BENVOLIO. I pray thee, good Mercutio, let's retire.
The day is hot, the Capulets abroad.
And if we meet, we shall not scape a brawl... (3.1.1-3)

There is a brawl, of course—which comes as no real surprise to anybody—in which Mercutio is killed by Tybalt, and Tybalt is killed by Romeo,

Tybalt seems to have foreshadowed this unfortunate turn of events at the Capulet's feast in act 1, scene 5, when Lord Capulet tells Tybalt simply to endure Romeo's presence, and not cause any trouble.

TYBALT. Patience perforce with wilful choler meeting
Makes my flesh tremble in their different greeting.
I will withdraw; but this intrusion shall,
Now seeming sweet, convert to bitt'rest gall. (1.5.94-97)

The confrontation between and among Tybalt, Mercutio, and Romeo was inevitable, and Tybalt simply foretold that something bad was going to happen, which it did. The audience might not have foreseen the extent to which something bad was going to happen, but they certainly knew that something bad would happen.

The situation is much the same as in Hamlet, after Hamlet tells Horatio what the ghost of his father said to him.

HAMLET. There's ne'er a villain dwelling in all Denmark
But he's an arrant knave.

HORATIO. There needs no ghost, my lord, come from the grave
To tell us this. (1.5.134-138)

Foreshadowing isn't foreshadowing if the audience knows what's going to happen, or if what happens—like a serious confrontation in the street between sword-carrying major characters from warring families—can easily be foreseen.

Shakespeare's use of foreshadowing in his play Romeo and Juliet could best be described as generous. It's never a surprise that the couple dies; it's even stated plainly as a fact in the prologue. The suspense in the narrative instead lies in how the tragedy comes to pass.

Other examples of foreshadowing in the play are too numerous to list here, but they include Juliet's claim that her "grave is like to be [her] wedding bed." In this quote, she is trying to say that she's worried she won't get to marry Romeo because he might already be married, but her words are also heavy with foreshadowing. Another, creepier example is when the lovers tell each other that they look pale. They think they're pale because of their sadness, but to the reader, this signals something much more tragic.

In Act 1, scene 4, Romeo's friends attempt to convince him to attend the Capulets' party with them.  He doesn't particularly want to go because he is feeling quite depressed as a result of his unrequited love for Rosaline.  By the end of the scene, however, they've convinced him to go and try to have some fun.  However, he says,

my mind misgives
Some consequence yet hanging in the stars
Shall bitterly begin his fearful date
With this night's revels, and expire the term
Of a despised life closed in my breast
By some vile forfeit of untimely death. 
But he that hath the steerage of my course
Direct my sail.  (1.4.113-120)

In other words, Romeo has a feeling that going to this party will be the beginning of a fateful chain of events that will end with his death.  However, whoever (or whatever) is in charge of his life's path is directing him to the party nonetheless.  Romeo's words foreshadow his eventual tragic death and the fact that the path to it does, indeed, begin tonight, when he meets Juliet Capulet.

Another example of foreshadowing comes when Romeo and Juliet are saying goodbye to one another after their one night together as a married couple.  Juliet says,

O God, I have an ill-divining soul!
Methinks I see thee, now thou are so low,
As one dead in the bottom of a tomb.  (3.5.54-56)

As Romeo climbs down from her window, Juliet feels as though her soul is predicting something terrible: with him below her, she feels as though he seems like a dead person at the bottom of his grave.  This foreshadows the fact that Juliet will never again see Romeo alive.  The next (and last) time she will see him is just after he has poisoned himself in her tomb, believing that she is dead.

Why did Shakespeare use foreshadowing in this line from "Romeo and Juliet"?

Shakespeare uses foreshadowing throughout Romeo and Juliet. The play repeatedly hints that it will end in death. Juliet repeatedly talks about death after having met Romeo.

The reason Shakespeare, or any author for that matter, uses foreshadowing is to build suspense and tension. It alerts the reader to what is coming and heightens our anticipation of that event. The line in question is an interesting bit of foreshadowing, because it doesn't focus on all of the misery that comes later in the play. The line actually displays a very positive outlook on the future. It hints that the marriage will happen and will end the feud between the two families. The line gives the reader hope that the play will end on a happy note.

In fact, all of it comes true. Romeo and Juliet do get married. The Capulets and Montagues do agree to end the feud.  

"A glooming peace this morning with it brings."

Unfortunately, Romeo and Juliet had to die to make the friar's quote come true—a tragedy hinted at by the description of the peace as "glooming."

How does the Prologue of Romeo and Juliet use foreshadowing?

The Prologue shows foresight and a vision of what is to come with its direct reference to Romeo and Juliet and the world in which they live.  Within this, Shakespeare is able to foreshadow one of his fundamental themes in the play that love amidst a world of hatred is possible.  The Prologue foreshadows violence between both families and the conditions of violence that exist in Verona.  Consider the following line:

From ancient grudge break to new mutiny,
Where civil blood makes civil hands unclean.

This foreshadows that this drama is one where violence is present.  The Shakespearean image of the bloody hand helps to bring this out in terms of the hands that will bring about violence to others.  At the same time, Shakespeare contrasts this world of bloodshed and violence with the tenderness of love, redemption being cast alongside despair:

A pair of star-cross'd lovers take their life;

This is a significant line because it foreshadows a couple of realities that will become clear in the play.  The first is that the couple is going against the natural order of violence and hatred that surrounds them.  It is for this reason that they are called "star- cross'd."  Additionally, the ending of the play is present in the idea of their death.  To the extent that Shakespeare "tips his hand" at the ending, he also reminds us that through this act, there will be reconciliation and redemption:

Doth, with their death, bury their parents’ strife.
The fearful passage of their death-mark'd love,
And the continuance of their parents’ rage,

This helps the audience understand that the solution to this condition of bloodshed is the presence of love, a theme that is foreshadowed here to be developed in the play.  The last foreshadowing element is actually a humorous one.  The Prologue reminds the audience that the play is two hours' "traffic" and that one needs to be ready in order to absorb that which is about to unfold.

How would you analyze the use of foreshadowing in dialogue in "Romeo and Juliet"?

I believe the statement: "Shakespeare uses foreshadowing in Romeo and Juliet that is conveyed in the dialogue" (I would eliminate the ubiquitous and redundant term "a lot" from this statement) is a good start. 

In Act I, Scene 1, Benvolio foreshadows Romeo's meeting with Juliet when he tells Romeo to look at other women. Romeo is sad and depressed over his unreciprocated love for Rosaline so Benvolio suggests Romeo "examine other beauties." And in Act I, Scene 2, after meeting the servingman and discovering Rosaline will be at Capulet's party, Benvolio tells Romeo,

At this same ancient feast of Capulet’s Sups the fair Rosaline whom thou so loves, With all the admirèd beauties of Verona. Go thither, and with unattainted eye Compare her face with some that I shall show, And I will make thee think thy swan a crow. 

Benvolio, of course, is right. Romeo falls madly in love with Juliet on first sight. He even echoes Benvolio's use of the word crow when comparing Juliet to the other women at the party. He says in Act I, Scene 5,

So shows a snowy dove trooping with crows

In Act I, Scene 4, Romeo foreshadows his own death as believes attending Capulet's party will lead to some fateful events which will change his life forever. He says,

I fear too early, for my mind misgives
Some consequence yet hanging in the stars
Shall bitterly begin his fearful date
With this night’s revels, and expire the term
Of a despisèd life closed in my breast
By some vile forfeit of untimely death.

Do thou but close our hands with holy words,
Then love-devouring death do what he dare,
It is enough I may but call her mine.