Oh my goodness! Please find another dietician because that's so ridiculous, it's scary that people with an education are giving out that kind of false information. There is no gluten in coffee cups or coffee, for that matter, so enjoy your take-out!
Some glues may have gluten in them but we are very sensitive and have had no problem with the cups. Some decaf coffees have gluten in them because of the processing though starbucks' does not. I would be more worried about the lining, if it is parafin or a similar petroleum product, you could have a problem with mineral oil allergies.