Keep in mind that the main thrust of Romeo and Juliet is not the forbidden love, but the senseless feud between the Montagues and Capulets. Pay close attention to the Prologue to Act 1:
Two households, both alike in dignity,
In fair Verona, where we lay our scene,
From ancient grudge break to new mutiny,
Where civil blood makes civil hands unclean.
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.
The fearful passage of their death-mark’d love,
And the continuance of their parents’ rage,
Which, but their children’s end, nought could remove,
Is now the two hours’ traffic of our stage;
The which if you with patient ears attend,
What here shall miss, our toil shall strive to mend.
At the end of the play, the Prince makes this blatantly clear:
Where be these enemies? Capulet! Montague!
See what a scourge is laid upon your hate,
That heaven finds means to kill your joys with love.
And I for winking at your discords too
Have lost a brace of kinsmen. All are punish’d. (V.3.296-300)
In other words, the feud itself is the problem being addressed; the death of the lovers is merely a symptom of that problem.
So yes, of course the play is relevant today, as it demonstrates the social poisons caused by meaningless hatred, as feuds tend to become over time. In the case of Romeo and Juliet, the Capulets and Montagues are fighting because of some "ancient grudge"; we aren't told why. We know, however, that their children--who certainly weren't around when the grudge began--and their servants hate the opposing houses, the opposing children, and the opposing servants. Again, none of them know why. They have merely inherited their hatred.
Consider, now, current situations in which ancient hatred is still inherited, where peoples hate and kill because "that's the way it's always been." Racism in America. Arabs and Jews. Yes...we have much to learn from this example of what comes of meaningless hatred.
Of course it is. Let me give you a few reasons why. There will always be forbidden love in our world. This is especially so, because our world is still very divided in terms of religion, race, and class.
For example, if a religious Jew wanted to marry a Gentile, there might be tension. If a devout Muslim wanted to marry a Christian, again there might be a tension. If a poor boy from the ghetto wanted to marry a rich girl from the upper crust of society, there might be strictures. As much as we like to think that the world has progressed, it has not. There will always be division and conflict. And whenever these divisions exist, love that crosses these divides may cause problems.
Second, as long as people fall in love, the play will have a foothold in our hearts. Love and passion of youth are things that we all experience or perhaps want to experience.
Finally, the play also celebrates what it means to be young. We will always have young people.
For all of these reasons, the Romeo and Juliet will live on and be relevant.
I agree that it is still relevant. The impulses of young lovers who still take dangerous risks are evident , even in this day and age. Youth cannot seem to see past the here and now, and when they are denied being together, they feed off each other, and many run away or take other rash actions all in the name of love. While the family feuds are less prevalent in this day and time, other factors still exist that prevent teenage couples from being allowed to be together.
My first impulse was to simply say, "Yes," but I'll resist that temptation. Yes, the play is still relevant. Some culture or period specific elements of it don't apply. We do not, for example, usually wall up our young women, or, as a rule, choose who they will marry (though that happens sometimes). However, the larger issues do still apply. There will always be love that is forbidden in some way—by parents, society, etc.—and there will always be young lovers who are electrified by one another's sight, touch, etc., and risk all for that love.