From the University of Chicago Celiac Disease Center:

Iron, calcium and Vitamin D are the most common deficiencies, but some present with deficiencies in B12, copper, folate, magnesium, niacin, riboflavin and/or zinc.  Nutrient deficiencies associated with celiac disease are due to intestinal damage.  In most cases, nutrient deficiencies that were caused by damage from celiac disease will naturally resolve as your intestine heals.  However, many gluten-free foods aren’t fortified with vitamins and minerals like their gluten-containing counterparts. Thus, we suggest a general multivitamin to prevent against nutritional deficiencies.