Police accused the Uamsho (Awakening) group of ordering its supporters on to the streets - but the movement's leadership denied having any part in the unrest.
Clashes broke out in the historic commercial and tourism centre of Stone Town on Saturday night and continued until Sunday afternoon, witnesses told Reuters.
The reports of attacks on churches will raise fears of an escalation in religious tensions in the predominantly Muslim island ruled by a semi-autonomous secular government. Several bars were attacked last year.
Shops remained closed and people stayed in their homes on Sunday as riot police patrolled the streets of Stone Town, witnesses said.
"There is a lot of tension, people are hiding in their houses. All the streets are empty, it's the only the police who are out there," resident Mai Zuberi told Reuters by phone.
"There were clashes all night on Saturday and the violence continued until noon today. All the roads were blocked in the Stone Town area," Zuberi added.