First-responders and fire brigades are attempting to control the blaze and maintain security.
On Tuesday, Prime Minister Haider al-Abadi warned that security agencies had evidence of "unprecedented" violations during the elections.
Though the cause of the fire was not reported, Mr Abadi described it as a "plot to harm the [Iraqi] nation and its democracy".
An interior ministry spokesman told Reuters news agency the blaze had been confined to one of four warehouses.
Iraq's parliament last week ordered a manual recount of the May 12 legislative elections and sacked the commission which oversaw the polls that resulted in a surprise victory for populist Shia cleric Muqtada al-Sadr.
The extent of the damage caused to ballot boxes was still unclear but some officials have suggested a lot of them had been spared. "Iraq is my concern, positions for me do not mean much", Sadr further said in the statement, urging the Iraqi nation not to pursue a recount. The announcement came a few days after Abadi ordered the creation of a high-powered commission to look into the alleged irregularities in the parliamentary elections.
The June 11 report said the suspects included three police officers and an employee of Iraq's elections commission.
His victory came on the back of an election campaign of opposing foreign interference in Iraq, as well as promising to rebuild schools and hospitals - which were heavily damaged in the war against the Islamic State of Iraq and the Levant (ISIL, known as ISIS) group past year.
The election's votes were initially counted electronically, and parliament has since called for a full recount. He was quoted saying, "The criminals manipulated the results of the elections because they know that the results that were announced are not true, so they burned the ballot boxes.This crime is not an ordinary crime and not a passing incident, but a major crime and a great betrayal of the country".
Less than half of Iraq's 24-million electorate took part, dumping the old guard in favor of Sadr's alliance followed by a list of former fighters of the Hashed al-Shaabi alliance that a year ago played a key role in the defeat of the Daesh terror group.
The move could undermine nationalist cleric Moqtada al-Sadr, who's bloc won the largest amount of seats in the elections.
"The Supreme Judicial Council's meeting saw the naming of the judges nominated to take over the duties of the board of commissioners of the Independent High Elections Commission", Judge Abdul-Sattar al-Birqdar said in a statement.
"We call for the election to be repeated", added Jabouri, who lost his seat in the election.
He said the main issue was with the electronic vote-counting machines that were used for the first time on 12 May.