Towards the end of part 3, while Scarlioni says that he will destroy Paris, the door behind him is open. The next camera shot of him has the door closed.
The Doctor marks six canvases so in the future they can be identified as the fakes, or copies of the original Mona Lisa. However, the Mona Lisa was painted on poplar, not canvas.
The Louvre seems incredibly accessible at night time, considering it houses such priceless works of art. The (off camera) window that was thin enough for Duggan to simply burst through to escape would never be in the real Louvre.
The Doctor (barely) thumps the guard (Tancredi's soldier) on the chin. Hitting him on the chin would not not be enough to knock him out. Had it been with any real force though, it could have knocked a few of his teeth out.
When the Doctor is dictating his letter to Leonardo, he says, "Sorry about the mess on the panels," but "canvasses" is written instead of "panels". This might have been an attempt to use ADR to correct the error of the the prop Mona Lisa's being shown on canvas instead of on wood panels like the real Mona Lisa.
The Count has Leonardo Da Vinci paint six additional copies of the Mona Lisa so that the Count can sell them all in hundreds of years time to fund his time travel experiments. There is no logical reason for such an elaborate and preposterous scheme. The Count could simply have bought land or shares in certain companies, or even brought about new inventions (since he had the technological knowledge, and already implied he invented the wheel and enabled the pyramids to be built) in order to make his future self richer.