The Muslim Brotherhood, whose leaders included Morsy before his election, has lost four members to violence in recent days.
On Sunday, a spokesman for the group said that its headquarters in Cairo fell under attack from between 100 and 150 "thugs." They threw stones and Molotov cocktails, spokesman Gihad Haddad said.
Police in charge of protecting the building abandoned it, he said.
Two people were shot dead Thursday when armed men attacked Muslim Brotherhood offices in Zagazig, Morsy's hometown, the spokesman said. The gunmen were shouting "Down with Morsy" during the assault.
State media reported that protesters demanding Morsy's ouster ransacked the city's offices of his Freedom and Justice Party, the Muslim Brotherhood's political wing.
Hamdeen Sabahy, a leftist former presidential candidate, said he and others challenging the Egyptian government "reject the violence ... and all the attacks on the (offices) of the Muslim Brotherhood or their party."
"Peace is our weapon," Sabahy said in a video message posted Saturday.
He urged his followers to continue their demonstrations nationwide, saying, "We are not fighting a rock, rather failing politics that do not meet the needs of the people."
Since Morsy took office, Egypt's already sour economy has plummeted further as investors pulled out of the country, and tourism has dropped.
At the same time, crime in Egypt has spiked. Some are calling for a return to the law and order they knew under Mubarak's autocratic rule, carried out with the iron hand of the military.
Although Morsy gave a speech highlighting his achievements during his first year in office, opposition members said he did not address his shortcomings and called for snap presidential polls, a new government and constitutional amendments.
Army allowed to crack down
Last week, Defense Minister Abdel-Fattah El-Sisi said the army would, if necessary, "prevent Egypt from slipping into a dark tunnel of civil unrest and killing, sectarianism and the collapse of state institutions."
His remarks raised the specter of a return to the powerful role the military played in domestic politics under Mubarak.
But Morsy is not Mubarak. He was democratically elected. That fact should be respected, his supporters say.
A history of violent crackdowns
In the past year, about 80 people have been killed during protests and other political violence largely attributed to Egyptian security forces, Amnesty International said.
The United States has approved the departure of embassy and consulate employees and their dependents because of the unrest, a senior State Department official said.
The State Department warned Americans to cancel all but essential travel to or within Egypt.
About 200 U.S. Marines in Sigonella, Italy, and Moron, Spain, have been put on alert as a precaution, according to two Obama administration officials.