BEIRUT—Like Israel and Saudi Arabia, Syrian President Bashar al-Assad has his own reasons to be worried about Iran’s framework nuclear agreement with the U.S. and other world powers.

On the face of it, the prospect of a final accord that would lift some international sanctions against Iran is good news for Mr. Assad and his regime, which depend on billions of dollars in support from Tehran. Add to that the direct and substantial military support Mr. Assad receives from Iran’s most powerful proxy force in the region, the Lebanese...