Statues are smashed using sledgehammers and drills in what seems to be a museum in the city of Mosul.
Statues are also shown being destroyed at an archaeological site known as the Nergal Gate.
World heritage body Unesco has called for an emergency meeting of the UN Security Council to discuss how to protect Iraq's cultural heritage.
In a statement, Unesco head Irina Bokova said: "This attack is far more than a cultural tragedy. This is also a security issue as it fuels sectarianism, violent extremism and conflict in Iraq."
In the video released via IS social media sites, black-clad men push over statues, smash them with sledgehammers and use a pneumatic drill to destroy the rubble.
The video shows a black-clad man drilling through and pulling apart what appears to be a stone winged-bull Assyrian protective deity that dates back to the 9th Century BC.
One of the militants in the video describes the artefacts as "false idols" and seeks to justify their destruction in religious terms.
Analysts say the artefacts are unique and priceless although the museum does also house copies of some items.
Ihsan Fethi, an Iraqi professor of architecture based in Amman, Jordan, told Agence France-Presse the destruction was "a terrible loss and an unbelievable act of cultural terrorism".
Amir al-Jumaili, a professor at the Archaeology College in Mosul, told Associated Press: "I'm totally shocked. It's a catastrophe. With the destruction of these artefacts, we can no longer be proud of Mosul's civilisation."
IS have controlled Mosul, Iraq's second largest city, since June 2014. The US military have said that an assault on the city by the Iraqi army could happen within months.
The region under IS control in Iraq has nearly 1,800 of Iraq's 12,000 registered archaeological sites.
The reported destruction of the statues follows recent reports that IS burnt down Mosul Library, which housed over 8,000 ancient manuscripts.
The Islamic State (IS) group has released a video appearing to show the destruction of statues in Iraq.