The main effect of the Boston Tea Party was that it angered the British and made them crack down even harder on the Patriots in Boston. This, in turn, made the Patriots even more angry at the British. So the longer term effect was that the Tea Party made the Revolution more likely.
The Tea Party really angered the British because it was an incident in which people broke the law and destroyed huge amounts of very valuable product. This would be like, for example, burning down some store's warehouse -- it's not a joking matter. When this happened, the British responded with the Coercive Acts, punishing Boston. As I said, this angered the Bostonians and made the Revolution more likely.
Expanding on the previous answer, remember that the Boston Tea Party was carried out by a handful of people, members of the radical Sons of Liberty, which most people in Boston disagreed with or at least dismissed. But instead of pursuing and punishing the few radicals, King George III and Parliament overreacted and punished the entire colony of Massachusetts.
The colonists at the time, both in Massachusetts and surrounding colonies, did not react to the Tea Party itself, just as they had not reacted to the Tea Act which supposedly motivated it, they remembered and hated the fact that the King was punishing his own citizens, wrongfully and without cause, for a crime they did not commit. This was what gained support for independence, and in a way the Sons of Liberty could not have dreamed of.