The real Orsino Orsini was Giulia Farnese's husband, not a cardinal. The Borgias did poison a Giambattista Orsini in 1503, not 1492.
While Pope Innocentius VIII lies in state, the Cardinals pay their respect by kissing his ring but in fact this very ring, the Fishersman Ring is destroyed by the Camerlengo of the Holy Roman Church upon the moment of determining the Pope's death in order to prevent it from being misused during these early hours of Sede Vacante.
Popes didn't wear white until the papacy of Pius V (1566-1572). He was a member of the Dominican Orders (Order of Preachers) and continued to wear the white habit of the order after he was elected pope. Previously, popes dressed in crimson like the cardinals along with certain vestments peculiar to their office.
The use of smoke to signal papal election results only dates back to 1870.
At the cardinals' banquet, one of them speaks this holy-sounding familiar phrase: "There go we, but for the grace of God." He is quoting John Bradford, who expressed this sentiment as he saw prisoners being led to their deaths during the reign of Queen Mary, about 60 years after the setting of this scene.
Cesare carries a Capuchin monkey to the dinner, a species native to Central and South America that in 1492 was still unknown in Europe (Columbus had only visited the Caribbean islands in that year, none of which have native monkeys).
The coronation anthem used is "Zadok The Priest". This was in fact composed in 1727 by Handel for George II of England's coronation, long after the historical events depicted.