DAMASCUS, Syria — A Jewish synagogue in Damascus believed to be thousands of years old has been damaged and looted as clashes have consumed the surrounding neighborhood, a Syrian official and an anti-government activist said Monday.

Damage to the Jobar Synagogue, which tradition holds was built by the biblical prophet Elisha, is the latest example of Syria’s rich cultural heritage falling victim to the civil war between President Bashar Assad’s regime and rebels seeking his ouster.

Syria is home to thousands of years of civilizations at the crossroads of the Levant and boasts important cultural sites dating back to the Bible, the ancient Roman empire, the Crusaders and the arrival of Islam....