No dogs die during the events of the movie but a character makes reference to his dog having been shot in London, presumably in an effort to reduce the spread of disease. The movie predates the Great Plague of London, in which this was a very widespread practice, but there are other anachronistic features in this film so it's not a stretch.
Except for some horses being ridden in the opening scene there are so animals in the film. It includes a "No animals were harmed" message in the credits anyway