Hakim opposed Saddam Hussein from exile in Iran for more than two decades, before returning to Iraq in 2003 after the US-led invasion.
He had chosen to receive chemotherapy for lung cancer in Iran, but his body is due to be transported to the Shia holy city of Najaf for burial.
Iranian and Iraqi mourners gathered as the coffin, draped in an Iraqi flag, was borne through a Tehran street.
He took control of the Supreme Council for the Islamic Revolution in Iraq (Sciri - which later became SIIC) after his brother was assassinated in Najaf in 2003.
The party has several senior cabinet members, and its militia - the Badr Brigade - has at times wielded considerable influence in Iraq's security establishment.
Since falling ill, Hakim had cut back his political involvement and his son Ammar gained prominence. He is expected to take over leadership of the party.
As heir to the leadership of one of the main anti-Saddam Hussein factions in Iraq, Abdul Aziz Hakim managed to keep good ties with both the American authorities and Iran, which strongly backed his group.
His brother and predecessor as party leader was the charismatic Ayatollah Muhammad Baqr al-Hakim, who was killed along with about 100 supporters in a massive car bombing in the city of Najaf in August 2003.
The family is revered among Iraq's largest religious community, the Shia, for its tradition of scholarship and its bouts of resistance against Saddam Hussein in its southern Iraqi stronghold.
However, the quietly-spoken Hakim was distrusted by many Sunnis who saw him as too Iranian-orientated and sectarian in his political philosophy.
In 2007, the party changed its name from Sciri - the Supreme Council for the Islamic Revolution in Iraq - to the Supreme Islamic Iraqi Council.
SIIC has been part of Iraq's ruling Shia alliance, the United Iraqi Alliance, led by the Islamic Dawa party of Prime Minister Nouri Maliki.
However, ahead of national elections in January, the SIIC announced last week that it would campaign from within a new Shia Muslim bloc.
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