A riot broke out over plans to convert a building in a Cairo suburb into a Christian church on Sunday.
Muslims and Christians clashed in Mataria after worshippers arrived for a service at the site of the planned church.
One man died in Aswan on the same day during a separate riot over the police killing of a suspected criminal. It is believed he died after inhaling tear gas.
Police in Mataria intervened when large numbers of Muslims and Christians faced off over a building which the Christians want to convert into a church. The police then clashed mainly with the Muslim side, they said.
In Aswan, about 450 miles to the south, riot police used tear gas against civilians protesting that a policeman shot dead a man in the town on Saturday.
Police said the man was a wanted criminal but the man's relatives said a police officer killed him in error and then the police tried to cover up the incident, the sources said.
Thousands of people attended the funeral on Sunday, some chanting: 'Illegitimate government, unjust government.'
In both cases people threw stones and bottles at the police. Two of the policemen were injured in Aswan and two in Cairo, and the protesters damaged shop fronts in both places, they said.
In the incident in Mataria, the confrontation between Muslims and Christians was the culmination of a long-running dispute over the plan to build a church there.
Residents said Christians had come from other parts of the city for the Sunday service at the building, which is not licensed as a church, and Muslims opposed to a church gathered against them. Some chanted: 'We're going to knock down the church' and shouted slogans of loyalty to Islam, witnesses said.
Relations between Egyptian Muslims and the Christian minority, estimated at up to 10 per cent of the population, are usually peaceful, but unrest breaks out from time to time over new churches, conversions and inter-confessional marriages.