At least two men died in overnight fighting in the canal city of Port Said in the latest outbreak of violence unleashed last week on the eve of the anniversary of the 2011 revolt that brought down dictator Hosni Mubarak.
Political opponents spurned a call by Mursi for talks on Monday to try to end the violence.
Instead, huge crowds of protesters took to the streets in Cairo, Alexandria and in the three Suez Canal cities – Port Said, Ismailia and Suez - where Mursi imposed emergency rule and a curfew on Sunday.
"Down, down with Mohamed Mursi! Down, down with the state of emergency!" crowds shouted in Ismailia. In Cairo, flames lit up the night sky as protesters set police vehicles ablaze.
In Port Said, men attacked police stations after dark. A security source said some police and troops were injured. A medical source said two men were killed and 12 injured in the clashes, including 10 with gunshot wounds.
"The people want to bring down the regime," crowds chanted in Alexandria. "Leave means go, and don't say no!"
The army has already been deployed in Port Said and Suez and the government agreed a measure to let soldiers arrest civilians as part of the state of emergency.
In Cairo on Monday, police fired volleys of teargas at stone-throwing protesters near Tahrir Square, cauldron of the anti-Mubarak uprising. Demonstrators stormed into the downtown Semiramis Intercontinental hotel and burned two police vehicles.
A 46-year-old bystander was killed by a gunshot early on Monday, a security source said. It was not clear who fired.
Mursi's invitation to opponents to hold a national dialogue on Monday was spurned by the main opposition National Salvation Front coalition, which rejected the offer as "cosmetic and not substantive".
The only liberal politician who attended, Ayman Nour, told Egypt's al-Hayat channel after the meeting ended late on Monday that attendees agreed to meet again in a week.
He said Mursi had promised to look at changes to the constitution requested by the opposition but did not consider the opposition's request for a government of national unity.
The president announced the emergency measures on television on Sunday: "The protection of the nation is the responsibility of everyone. We will confront any threat to its security with force and firmness within the remit of the law," Mursi said.