Location, Location, Location
I saw an article on this subject on The Daily Telepgraph website, entitled "The Duchess: On set in the Peak District".
The main claim to fame for The Three Cups hotel as a film location is "The French Lieutenant's Woman" from 1981. That film tells a rather sad story of relationships which went wrong and, from the reviews of The Duchess film, it sounds like there is much sadness in the new film.
Cultural tourists looking for something more uplifting may like to visit Lyme Regis and think of its literary connections with Tolkien, Chesterton and Tennyson who all stayed at The Three Cups. They may also enjoy the associations with Jane Austen.