Mr Trump made the threats amid fallout from the US assassination of Iranian commander Qasem Soleimani.
The president said cultural sites were among 52 identified Iranian targets that could be attacked if Iranians “torture, maim and blow up our people”.
But the UN’s cultural organisation and UK foreign secretary were among those to note that such sites were protected.
The US and Iran have signed conventions to protect cultural heritage, including during conflict. Military attacks targeting cultural sites are considered war crimes under international law.
Qasem Soleimani was killed in a US drone strike in Baghdad on Friday on the orders of Mr Trump. The killing has sharply increased regional tensions, with Iran threatening “severe revenge”.