Staff at the consulate had evacuated before the incident, they said. Authorities announced a curfew shortly afterwards, state media reported.
The protesters entered the compound late on Wednesday and set fire to the entire consulate building, the police and civil defence source said.
Iraqis have taken to the street in their thousands to demand the departure of their government and political class which they view as corrupt and beholden to foreign powers, especially neighbouring Iran.
Authorities have shot dead more than 300 people in an attempted crackdown on the unrest, according to police and medics.
Iranian-backed parties and paramilitary groups dominate state institutions and parliament. (Reporting by John Davison and Alaa Swilam)
