Egypt's PM has appealed for calm after 24 people were killed as clashes between Coptic Christians and security forces escalated into full-scale riots.
The violence broke out after a protest in Cairo against an attack on a church in Aswan province last week.
Some Muslims joined the Copts in protesting against military rule while others responded to government calls to help preserve stability.
An emergency cabinet meeting has been called for Monday.
The BBC's Yolande Knell in Cairo says there is pressure on ministers and on the country's military rulers to give assurances about national unity.
A nighttime curfew was lifted at 07:00 local time (05:00 GMT).
Mr Sharaf, who toured the area where the clashes occurred, also addressed calls by protesters for the removal of the military rulers.
"The most serious threat to the country's security is tampering with national unity, and the stirring of discord between Muslim and Christian sons of Egypt," he said in a televised address late on Sunday.
He added that such violence - the worst in Egypt since President Hosni Mubarak was ousted in February - was "tampering with the relationship between the people and the army".
Earlier, Egyptian TV showed protesters clashing with security forces as army vehicles burned outside the state TV building where protesters had originally planned a sit-in.
There were also reports of burning vehicles outside the Coptic hospital, where many of the injured have been taken.
Sectarian tensions have increased in recent months in Egypt.
The Copts - who make up about 10% of the population - accuse the governing military council of being too lenient on the perpetrators of a string of anti-Christian attacks.
Thousands - mainly but not exclusively Christians - joined the initial march from the Shubra district of northern Cairo to the state TV building in Maspero Square.
They were calling on the military council to sack the governor of Aswan province. They also accused state TV of fanning the flames of anti-Christian agitation.
But the demonstrators said they were assaulted by attackers in plain clothes before the clashes with the security forces broke out.
The violence began outside the state TV building but soon spread to Tahrir Square, the epicentre of the demonstrations which led to President Mubarak's resignation.
There were reports of thousands joining in the street violence, attacking both sides. Rioters tore up the pavement and hurled stones.
Correspondents say that many Muslims came out to defend Christians from the security forces and protest against the military's continued hold on power.
Some called for the resignation of the military council, in particular its chairman, Field Marshal Mohamed Tantawi.
Others responded to government calls to help the army quell the unrest for the sake of stability.
Some protesters reported hearing gunfire, and several said they had seen a military vehicle run over at least five people.
Eyewitness Nigel Hetherington says troops fired rubber bullets and teargas into crowds. "I saw civilians running past my window as troops fired wildly into the crowds," he told the BBC.
Nazly Hussein, who was near the Ramses Hilton hotel, said she had not seen so many dead in the streets since the days between 28 January and 2 February, the most violent days of the revolution.
Another eyewitness, a Christian named Sandra, watched the violence on Maspero Square from home.
"Everything took place directly underneath our house," she told the BBC. "It all happened so fast. The army moved in with tanks and were literally running over people. You could see bodies all over the place. People were running and screaming."
Hossam Baghgat, head of the Egyptian Initiative for Human Rights and an eyewitness, described to local TV what he saw: "Some bodies had no heads, others had their heads completely flattened [and] you can't see the features, it's so obvious they were run over."
The authorities have so far made no comment on the reports of military vehicles running people over.
The ministry of health said that at least 24 people had been killed and 212 had been wounded in the violence.
Of this number, at least 86 were security forces, ministry spokesman Hisham Shiha told the BBC.
Our correspondent says sectarian tensions have simmered in the political and security vacuum that has developed in the past couple of months.
Christians have been worried by the increasing show of strength by ultra-conservative Islamists.
In May, 12 people died in attacks on Coptic churches. In March, 13 people were killed in clashes between Muslims and Copts in Tahrir Square.
This latest violence comes ahead of parliamentary elections scheduled for 28 November, the first such vote since former President Hosni Mubarak stepped down.
The Copts, the largest minority in Egypt, complain of discrimination, including a law requiring presidential permission for churches to be built. Egypt only recognises conversions from Christianity to Islam, not the other way.