On the opening shot of Edinburgh Castle, you can clearly see the spire of The Hub to the left of the screen. The Hub was not built until 1845. Also, The Hub is on the other side of the Castle from there.
Two hangings are shown in the film, in the style of a "long drop" designed to break the neck and cause instant death. This style was first introduced in the 1870's by William Marwood as a more humane approach. Before then, and certainly in 1828-1829 when the film was set, the victim was simply suspended by a rope around the neck, and choked to death by suffocation.
When Burke and Hare stop for a drink and Burke says that they have no money, Hare plucks a half-crown from underneath Burke's cap. The half-crown that is used in the film is from 1902 - 1910 which is nearly 100 years after the film is set.
When Burke & Hare chop down the tree to stop the coach, they hit the tree with their axes at chest height. When the coach approaches they have hardly made a dent in the tree and when they push to tip the tree over, it doesn't break at chest height, but the entire tree falls, arguably still fully intact.
In the "multiple dissection" scene Tom Wilkinson uncovers each of the four dead bodies one at a time. But, following a quick edit to the applauding students, in a closer shot one of the bodies is covered up again.
Near the end of the movie there is a high shot of Market Square, which is supposed to be below Edinburgh Castle as established during the opening scene of the movie. However the view looks almost identical to the Royal Mile which runs from the Castle itself down to Holyrood Palace. You can even see the spire from St Giles Cathedral in the background, which is on the Royal Mile.