Disposable paper cups are wax-coated to make them impervious for holding hot and cold liquids. I sincerely doubt you can 'taste' the cardboard.
You can encourage customers to Bring-Your-Own cup for sampling, if taste, paper waste and cup cost are an issue for your store or the customer. Alternatively, you can sell, for a nominal cost, logo-ed ceramic cups, or 'to-go' thermo-mugs - as JB suggests, above.
Samples are a vital part of specialty food and beverage merchandising. Even large grocery stores are finding that when they offer customers samples of show-cased items, it pays off handsomely in sales.
Tea sampling has an additional bonus - demonstration of suggested brewing technique or correction of brewing misconceptions. Can't stress the importance of this concept, else the customer may find that the tea tastes quite different when prepared haphazardly at home.
Your store has an advantage over the Adagio online retail model - you can test out new teas directly with customers. These teas would not be available through the corporate website unless selected later for national sales.
That's the hook - you can buy certain teas that are seasonal or special limited offerings at the shop. It keeps your product repertoire 'fresh' and interesting, and provides a reason for regular visits.