The compound in Baghdad was attacked by demonstrators furious over the deaths of militia members in US air strikes.
Mr Trump tweeted “Iran will be held fully responsible for lives lost, or damage incurred, at any of our facilities. They will pay a very BIG PRICE! This is not a Warning. It is a Threat. Happy New Year!
But Iran’s Supreme Leader Ayatollah Ali Khamenei responded by saying the US “can’t do a damn thing”.
Anti-American sentiment was widespread in Iraq, he added.
Ayatollah Khamenei also threatened the USA by tweeting “If the Islamic Republic decides to challenge & fight, it will do so unequivocally. We’re not after wars, but we strongly defend the Iranian nation’s interests, dignity, & glory. If anyone threatens that, we will unhesitatingly confront & strike them
