What is the dramatic significance of the deaths in Romeo and Juliet?

There is significance behind all of the deaths in Romeo and Juliet; each one leading to the end of the family feud between the Montagues and Capulets.
Mercutio (Romeo's best friend and heir of the Prince) dies standing up for Romeo. Mercutio acted impulsively in fighting Tybalt, and his death not only spared Romeo for the time being, but it also foreshadowed what was to come for the two young lovers (Romeo and Juliet) later in the play when his last words spoken placed a curse on the two families.
Tybalt's death (by the hand of Romeo) proved his love of family honor and how he would do anything to defend the Capulet name.
Some may say that Romeo and Juliet's deaths were immature, but they chose to take their lives to honor their love for one another, even at such a young age. Their suicides led to an end of the family feud. Without them dying for each other, the feud would still be going on, and others may have suffered because of it.

The dramatic significance of death in this play is that it is the one event that finally puts an end to the years-long feud between the Capulet and Montague families. The feud is what prompted the conflict of the play, what problematized the relationship between Romeo and Juliet from the start, what fueled Tybalt's rage and compelled him to defend his family's honor by challenging Romeo to a sword fight after Romeo crashed their party, and so on. Upon learning of their children's love for one another and their suicides, Lords Capulet and Montague finally put aside their differences; it took these deaths in order to make it happen. Capulet says,

"O brother Montague, give my thy hand.This is my daughter's jointure, for no moreCan I demand" (5.3.306-308).

Because there can be no celebration of the of the marriage and no gifts exchanged between the families, Capulet says that Montague's hand--a symbol for peace between them--can be all they have.
Montague responds,

"But I can give thee more,For I will ray her statue in pure gold,That whiles Verona by that name is known,There shall no figure at such rate be setAs that of true and faithful Juliet" (5.3.309-313).

Montague, then, vows to commission a golden statue of Juliet that will stand for centuries as a testament to Juliet's loyalty and fidelity. Thus, the dramatic significance of death in the play is that it is the only thing that compels the feuding families to put their antagonism behind them.