yes, and no.
I've thought about this a great deal actually, and the main concern is the location of the operation and the types of tea you are hoping to sell/promote. I'm assuming that the proprietor of the teashop is hoping to make his/her living by the business. If they have other means of income, they can make their tea shop as boutique and exclusive as they'd like.
In the United States, unless you are in a major metropolitan market, you'd be foolish not to offer coffee in my opinion. No matter how passionate you are about tea, in business you are selling sprockets and widgets. That's it. And you have to sell enough of them to stay in business, because if you don't, who will know about the great tea you have anyway? I think offering coffee will at least allow someone not currently in to tea to experience the store and pique their curiosity. If the store is in a metropolitan market with diverse, educated, and cultured potential customers to pull from, a strictly tea operation would probably do fine. There are plenty of examples to site. But, try starting a gong fu and/or Japanese focused tea store in the rural South, and well, I'd be skeptical at best.
I'm not saying it can't be done, I just think that in a predominately coffee drinking nation, you'd do well to diversify your offering from a brick and mortar store location. But I don't think anyone could survive these days without a substantial internet business for their tea shop.
I have never owned a goods business, so don't take my opinion too seriously. Where there's a will, there's a way...I guess.