BEIRUT—Islamic State attacks on Syria’s Assyrian population have driven the Christian minority into Lebanon, further straining the impoverished nation and raising concerns that Christianity is being depleted in the Middle East as refugees move on to Europe.

The Lebanese government, which early this year tightened visa regulations in a bid to stem the tide of Syrian refugees, this month granted the Assyrians an exception, citing the Islamic State campaign that began in late February. More than 1.2 million Syrians have overwhelmed...