MOSCOW—Russian authorities have ordered dating app Tinder to share user data and messages with government and intelligence agencies, the latest sign of Moscow’s tightening grip over online activities in the country and the Kremlin’s infringement on privacy.

Russia’s communications censor, Roskomnadzor, has put Tinder on a list of services required to store user data and communication and hand it over to the government, the agency said on Monday. Apps must also be ready to pass encryption keys to security agencies, including...