I wouldnt say domino's would have a issue with the piercing, he would be asked to cover it up with skin-coloured tape or bandaid, or use clear, plastic jewelery. Same goes for most public schools.
I would let him, more the eyebrow than the lip, as i personally think eyebrow piercings dont stand out as much as a lip piercing, and looks a bit nicer.
Word of warning with eyebrow piercings: My bro had one, and ripped it out while drying his hair with a towel. Not good lol.