Crowds in the streets of the Shia Muslim-majority city chanted against Iranian influence on Iraqi politics.
A protester was killed and 11 more wounded during clashes with security forces.
Friday saw a fifth day of violent protests in the country's second city, with at least 10 people killed.
Officials announced a curfew at about 21:00 local time (18:00 GMT).
Iran, the region's main Shia power, has fostered ties with Iraq's Shia majority since Saddam Hussein's downfall.
But the Iranian consulate is just another public building protesters associate with the government in Baghdad, analysts say.
Iran's foreign ministry spokesman, Bahram Qassemi, condemned the "agitated" attack, saying it had caused "considerable material damage".
All embassy staff are thought to have left the building before it was torched, Iranian media report.
In Thursday's protests, demonstrators burned local government buildings and political offices and forced the closure of Umm Qasr, the country's main sea port to the south of Basra.
On Friday, Iraqi Prime Minister Haider al-Abadi said he would speed up the release of funds intended to improve basic services in Basra.
Iraq's most senior Shia figure, Ayatollah Ali Sistani, blamed the unrest on political leaders and called for a new government "different from its predecessors".
Anger in Basra has grown over the lack of jobs, electricity and safe drinking water, with hundreds taken to hospital after drinking contaminated water.
Local residents say the government is corrupt and has allowed infrastructure to virtually collapse in the region that generates much of Iraq's oil wealth.
More about: Iran