The series opens with a young Da Vinci smoking a blend of tobacco and other herbs from a hookah. However Da Vinci was born in 1452, and tobacco wasn't imported from America to Europe until quite some time after 1492.
The show begins with the mystic Telling Da Vinci that the pipe contains a mix of tobacco and black hellbore. Tobacco is indigenous to the Americas and was not introduced to Europe until after Da Vinci's death.
(at around 20 mins) While Da Vinci is watching some birds at the Market, we can see that the local guardsmen wear the characteristic hat of the Spanish Civil Guard, which wasn't created until the late 19th Century.
Da Vinci is shown smoking tobacco. Smoking of tobacco was not possible in Europe at this (pre Columbian) time as tobacco is indigenous to the Americas and only came into use in Europe in the 17th century.
The name Vanessa (Leonardo's art model) was invented by Jonathan Swift in the early 1700s.
When DaVinci looks at the birds in the market, one of the birds shown is a Rainbow Lorikeet from Australia. Australia had not been discovered in DaVinci's lifetime.
The city guards wear the distinctive hat of present day Spanish Civil Guard. This hat is the evolution of the typical three-pointed hat typical of the XVIII century, and would be unthinkable in XV century Florence.
(at around 22 mins) There appear to be what look like 2 modern extending jib cranes, these don't appear in any of Leonardo Da Vinci's mechanical engineering sketches, so unlikely to be attributable to the great man.