The Tigers did not take anyone in the major league portion of the draft. They did lose RJ Petit to Colorado with the #1 overall pick.
In the minor league portion, the Tigers lost Austin Murr, Archer Brookman, Danny Ceretti, and Travis Kuhn.
Also in the minor league portion, they picked up:
RHP Luke Taggart from COL
RHP Jan Caraballo from SF
RHP John Stankiewicz from MIN
RHP Yendy Gomez from LAA
I'd never heard of any of the guys we acquired, including the player from the Giants.
My understanding is the minor league part of the draft is limited to AAA players and teams pay to make a selection but there are no roster restrictions like there are for players selected in the major league portion.