BEIRUT—Islamic State has mounted a propaganda blitz to discourage Muslims from fleeing Iraq and Syria for Europe, an exodus that threatens to weaken its efforts to recruit foreigners and deprive it of the skilled professionals needed to run its self-described caliphate.

Since mid-September, Islamic State has produced 14 videos and published some 17 articles in response to the waves of migrants and refugees heading to Europe. The flurry is unusual even for so prolific a propaganda machine as Islamic State’s, and underlines...