To remove your name from a Facebook photo that you've been tagged in, click remove tag next to your name in the image caption. screenshot by Dennis O'Reilly/CNET. You can tag a photo with the names of people who are not on your Friends list.o remove your name from a Facebook photo that you've been tagged in, click remove tag next to your name in the image caption. screenshot by Dennis O'Reilly/CNET. You can tag a photo with the names of people who are not on your Friends list.