What role does the feud between the families play in this tragedy?

The feud between the families is introduced at the very beginning of the play. In the opening prologue, we are told of "two households, both alike in dignity . . . From ancient grudge break to new mutiny." The fact that these are the very first words of the play indicates that this feud will be central to the story that unfolds and also that the feud is the cause of the events that follow. To compound this point, act one opens with a "civil brawl" between various members of the two families.
Romeo and Juliet's love throughout the play is always set against the background of the feud between their respective families. As a result, their love becomes more precarious, and it's all the more precious precisely because it is forbidden.
It's also important to note that nobody from either family seems to know or make reference to the original cause of the "ancient grudge." The families feud simply because they have always feuded. Thus, the fact that Romeo and Juliet's love comes to a tragic end because of the feud is all the more tragic because the feud continues for no good reason. Romeo and Juliet come to a tragic end for no greater reason than the surnames they bear. Indeed, in act 2, scene 2, Juliet asks, "What's Montague? What's in a name?" In these questions, Juliet echoes the incredulity of the audience.
Arguably, Friar Laurence's decision to marry Romeo and Juliet hastens the tragic outcome of the play. As a responsible figure of authority, he perhaps should have advised them more strongly to practice patience. Juliet later decides to take the potion because she can't bear the prospect of defiling her marriage to Romeo by marrying Paris. When Friar Lawrence decides to marry Romeo and Juliet, he does so because he hopes, naïvely, that their marriage might end the feud. In his own words, he hopes that their "alliance may so happy prove, / To turn your households' rancour to pure love." Thus the feud, in this instance, indirectly contributes to the tragedy of the play, as it leads Friar Lawrence to make a hasty, ill-fated decision.

The Feud Between the Capulets and Montagues  is the reason the tragedy between Romeo and Juliet even occurred. The two families have had a long feud with each other for years. Both Romeo and Juliet were forced to hide the feelings they had for each other, because of their families. 
If the Capulets and Montagues had resolved their differences, both Romeo and Juliet would have lived to have a happy life. The Montagues and Capulets realize the error of their ways a little too late. Nevertheless, both families resolve their differences after losing their loved ones.
 

The family feud between the Montagues and the Capulets is the foundation upon which this whole tragedy sits. Simply, without the feud, there would be no need for Romeo and Juliet to hide their love and to plot to be together, and it is likely that their unnecessary deaths would not have happened.
Interestingly, however, without the family feud and the excitement of forbidden love, would the attraction between Romeo and Juliet been as powerful? With this question in mind, you could argue that the family feud is actually necessary for their love affair to carry on for as long as it did and for as intensely as it did. Romeo, who is notoriously in love with love, may have lost interest had Juliet not been so out of reach.