Even after a celiac patient adopts a gluten-free diet, various vitamin deficiencies can persist even for years. At Beth Israel Deaconess Medical Center in Boston in 2009, researchers studied 109 men and women with celiac disease to see whether the food they were eating on the gluten-free diet was providing the daily recommended intakes for various nutrients. When the celiac patients recorded everything they ate over a three-day period, it turned out that many were not getting enough nutrients in their food. (For more information about this study, see Research “Supports the Recommendation for a Daily Multivitamin” for People with Celiac Disease.)
So the answer is that people with celiac disease usually do need supplemental vitamins. But as the Celiac Sprue Association cautions on its website, patients with celiac disease should not take nutrient supplements without a physician's monitoring. Current guidelines and recommendations for appropriate vitamin intakes are based on studies of healthy people without diseases. Patients with celiac disease should have their vitamin status analyzed by a registered dietitian and/or a physician in order to determine what vitamins they need to take, and how much.
Sources:
Clayton PT. B6-responsive disorders: a model of vitamin dependency. Journal of Inherited Metabolic Diseases 2006; 29: 317-26.
Kapur G et al. Iron supplementation in children with celiac disease. Indian Journal of Pediatrics 2003; 70: 955-8.

