While in the dinning room, the shadows on the painting of the horse change dramatically while Turner is singing and talking with the pianist.
Turner spits on his painting as he is working on it and continues painting. This is not possible. Oil and water (saliva) do not mix and the paint would turn into a balled up mess.
When Turner is tied to the mast of the ship the snowflakes stick to his face and are foam. Snow is not made up of foam.
In one of the first outdoor scenes of a street, two extras dressed in period costume can be seen stepping over a very modern looking BT manhole cover in the pavement.
When Dr Price attends Turner in London, he says he is taking the train back to Margate from Charing Cross station, which wasn't opened until 1864, 13 years after Turner's death.
When Turner says "no good deed goes unpunished" he's a bit ahead of his time. The quote is attributed to Clare Boothe Luce with some unsupported claims it might have first been said by 3 others, who all would have been quite young or unborn at the time of Turner's death.
In one scene, Queen Victoria is with Albert looking at paintings. She is wearing mauve, which she didn't wear until after his death.
When Mr. Turner is shown in the marketplace towards the beginning of the movie, a woman enters the scene carrying a pug, with a short snout which only appeared due to breeding after 1900s. Pugs back then had longer snouts.