I've run live vocals through a Tubescreamer and a Dano Daddy-O...never a DS1...I wonder if Distortion would be significantly different than an OD pedal...but I don't think it'll hurt the mic or the recorder...the real question is whether or not you get the sound you want. Usually I add distortion AFTER I've recorded a vocal track...there's some plugs that do it, but if I had to I'd run an out of my Delta 1010 into an OD pedal, and then back in again. Post a clip!
Jacob