"Are" is the form of "to be" that you use in a question. To ask "are you X?" is the same as to ask "If you said 'I am X' then would that be true?" "Do you X?" is the same as "Are you in the habit of doing X?" So hopefully it's clear that "Are you obsessed with grammar?" is the same as "If you said 'I am obsessed with grammar' then would that be true?" "Do you obsess with grammar?" is the same as "Are you in the habit of obsessing with grammar" which makes no practical sense (although it'd be fun to try and make it makes sense by pretending that you and someone called 'grammar' sit around and obsess in the abstract).

Hey! I think "are you obsessed with grammar" is the correct one. Because are is usually used for something that you have and do is more about things that you do. So we say are you a happy person and we say do you go to the gym.