How important is two weeks notice?

I'm recently turned twenty and in the beginning of August I'm going to be moving across the country with my girlfriend. My boss is the kind of guy who calls your parents or husband if there's any problem no matter your age, and I know he'll call my parents when I put my two week's notice in. I'm getting about seven to ten hours a week for no real reason (I'm a good employee and the managers are all asking for more hours for me to no avail) so if I did walk out, the schedule wouldn't get that messed up (I work at a fast food place). I know the managers would give me good references no matter what because I work hard, am never late, never call in sick, and I get along with everyone there. So....how important is a two week's notice? I can't let my parents know that I'm leaving until I'm actually gone because they're homophobic bigots and have threatened to kick me out if I became a lesbian before, and for now I'm dependent on them. It's a complicated situation.