Maamoun Abdulkarim said IS jihadists blew up the tombs of Mohammed bin Ali, a descendant of the Prophet Mohammed's cousin, and Nizar Abu Bahaaeddine, a religious figure from Palmyra, three days ago.
Bin Ali's burial place is located in a mountainous region four kilometres (almost three miles) north of Palmyra, in central Syria.
Photos published by IS depicted two armed men carrying cannisters, apparently filled with explosives, walking up the rocky hill to the site.
Abu Bahaaeddine's tomb, nestled in a leafy oasis about 500 metres (yards) from Palmyra's ancient ruins, is said to be more than five centuries old.
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"They consider these Islamic mausoleums to be against their beliefs, and they ban all visits to these sites," Abulkarim said.
Ten days ago, fighters from the jihadist group also destroyed a number of tombstones at a cemetery for Palmyra residents, Abulkarim told AFP.
"All tombs with marble designs were destroyed. For them, graves should not be visible," he said.
The extremist group captured Palmyra, a renowned UNESCO World Heritage site, from pro-government forces on May 21.
Syria's army has advanced in recent days west of the city, near key oil and gas fields.