Long Island is about 1200 square miles, just counting Nassau and Suffolk; saying what region is convenient for you would help with recommendations--telling you about a nice burger place in Wantagh when you live in Riverhead, or a place in Sayville if you live in Great Neck, is less than helpful.
Also helpful is to also put the "Long Island" part in the subject so that people don't skip over the thread because (since this board is also for upstaters) they don't know if the topic is local to them or not.
What sort of burger--fast-food style thin patty, or a thick one? Do you want "gourmet" toppings? What sort of ambiance? Alcohol available? Table or counter service? Should other items of at least decent quality be available?