Before the parliament had acted the electoral commission said it was voiding 1,021 ballot boxes from around the country, along with votes cast by Iraqis overseas and Iraqis still living in displacement camps that were set up during the battles against ISIL.
First-responders and fire brigades are attempting to control the blaze and maintain security.
Iraqi Prime Minister Haider al-Abadi called the fire "a plot to harm the nation and its democracy".
"We will take all necessary measures and strike with an iron fist all who undermine the security of the nation and its citizens".
The site was divided into four warehouses, said Interior Ministry spokesman Maj.
The authorities say the ballot boxes were saved and the blaze, whose cause is now under investigation, will not affect the recount.
But the result was contested following allegations of fraud namely by the veteran politicians, including parliamentary speaker Salim al-Juburi.
The election was won by Sadr's electoral alliance with communists, with long-time political figures pushed out by Iraqi voters who were hoping for change.
Cleric Moqtada al-Sadr called for Iraqis to show unity rather than squabble over a possible rerun of the election his bloc won last month, in remarks that seemed aimed at defusing political tension after a storage site holding ballot boxes caught fire.
The extent of the damage caused to ballot boxes was still unclear but some officials have suggested that a lot of them had been spared.
However, Baghdad province officials said that the boxes were part of a manual vote recount and had all been scorched.
Last week, Iraq's parliament voted to dismiss the commissioners and replace them with judges while calling for a full hand recount of the approximately 11 million votes.
Last month's election saw a record number of abstentions as Iraqis snubbed the corruption-tainted elite who have dominated the country since the 2003 US -led invasion that toppled Saddam Hussein.
The fire occurred on the eve of declared by the Parliament of the recount of votes in the elections, marked by numerous fraud and abuse.
Saeed Kakei, one of the members of the elections commission who himself produced what he said was evidence of fraud, also alleged the fire appeared to be deliberate and suggested significant violations and that "some individuals don't want a manual recount to happen", WSJ reported.
He said the main issue was with the electronic vote-counting machines that were used for the first time on 12 May.
The Baghdad warehouse complex where ballot boxes were stored.
Election victor Sadr on Monday called on Iraqis to unite rather than fight over the recount in a message aimed at unity following the fire.
But a repeat of the election is unlikely, analysts say, as none of the top parties have endorsed this step, and many incumbent lawmakers have lost their seats and thus lack legitimacy in the eyes of the public.
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