When a deck is completely milled and a player has no more cards in their library, if you make them draw a card will that make them lose?
Yes yes it would
120.4. (http://imtgapp.com/forum/index.php?action=imtg;area=rule;number=120.4.): A player who attempts to draw a card from a library with no cards in it loses the game the next time a player would receive priority. (This is a state-based action. See rule 704.)