The act of being forced to sacrifice time in order to hang out with either a girlfriend or a girl you are trying to bone. DAB is an acronym for Dumb Ass Bitch, so really, it is to say you are being sucked away from your friends or your responsibilities to hang out with some DAB, hence the expression, DAB Duty.
Jeremy: Chris, come play basketball with us.
Chris: Man...I WANT to, but I have to hang out with Chantel tonight.
Jeremy: What!? You're on DAB Duty AGAIN!? You're such a fag.