Man, no. Take the summer off. Go to the beach. Watch some television. Hang out with friends. Do the things you're not going to have time to do once school starts. Dental school's enough work as it is without getting all competitive before the school year's even begun. Once you're in, you're going to be absolutely inundated with information at every turn and breaks will be few/far between. Studying now is not going to help you in the slightest and, if anything, it's just going to hurt you. Enjoy your leisure time while you still have it.
((Protip: You absolutely can burn out from studying, even if you enjoy it, and at some point you probably will. Don't make that day come any faster than it already is.))
Second point of note, never think of your classmates as competition. That mindset is going to make you miserable in the long run. Your classmates have the potential to be the single largest and most helpful resource available to you over the course of the next few years. Working with them would be in your best interest.