No. Gluten-containing skin care products and cosmetics aren't a problem unless you accidentally swallow them. For this reason, avoid using such products on your lips or around your mouth. Also, avoid using gluten-containing dental products, such as certain mouthwashes and toothpastes. If you're uncertain about whether a product contains gluten, check the ingredient list on the product label or contact the manufacturer.
Some people develop a form of celiac disease called dermatitis herpetiformis (DH), which causes an itchy, blistering rash. This skin disorder is also linked to gluten intolerance. But although it involves the skin, DH is caused by ingesting gluten, not by skin contact with gluten. So, eliminating gluten from your diet will help clear up DH as well.
If you use a cosmetic or skin care product that contains gluten and you develop a skin reaction, see your doctor or dermatologist to identify the cause. It is possible to have an allergy to wheat or another grain that could cause a skin reaction.