When Rochester returns home, the carriage gets stuck in the mud so he gets out and walks towards the house. He's met by his wife. Rochester says, "Madam. This driveway will not do." The word "driveway" didn't come into use until around 1865 (according to dictionary.com) - about two hundred years after Rochester's death.
Looks closely at the back of the person being prodded and whipped around 1:17:10 (in Dr. Bendo's room shortly after Alcock pees into a vial), you can faintly see quite a large tattoo in the small of the lady's back that hasn't completely been covered by the make-up.
The Earl of Rochester reads an insulting poem he wrote about Charles II which implies that the King is impotent, and insists this "is true." In fact, Charles II was a noted womanizer who fathered at least a dozen illegitimate children by seven mistresses.
When the dog poops on the floor behind the king, the sunlight is shining brightly onto the floor. The camera cuts away to Chaffinch and when it returns to the king, there is no sunlight and the pile of dog poop has disappeared.
In the playhouse, after Harris announces the kings arrival, Lord Rochester starts to remove his hat with his right hand on the crown of the hat. In the next shot, he is removing it with his left hand on the brim of the hat.
The prologue and epilogue have the Earl-as-Narrator mentioning "gonads" (a word not coined until almost 1880), and remembering the circumstances of his own death. This part of the movie is meant to break the fourth wall. The narrating Earl knows he is speaking to a 21st-century audience.
In several shots, the fake nose John Malkovich is wearing is noticeable, including the sun glowing through when outdoors, and when in the theater box speaking to Johnny Depp, there is a distinct color mismatch that shows the outline of the prosthetic.