A gunfight broke out on Monday after protesters from Fatah threw stones at the house of a Hamas minister and chanted anti-government slogans in Bait Hanun in the northern Gaza Strip.
There were no reports of any casualties.
Another gunfight broke out at the main hospital in Gaza City when the family of one of those killed arrived to retrieve his body.
Gunmen from Mahmoud Abbas' Fatah faction accompanying the family opened fire on Hamas militiamen patrolling the hospital, sending patients and doctors running for cover. No one was injured in the violence, hospital officials said.
Fatah gunmen also shot at the deputy prime minister's bodyguards as they rode in a government car, injuring two of them. Nasser Shaer was not with them at the time of the attack. Hospital officials said a Fatah supporter was also injured in the fighting.
Hamas-controlled interior ministry forces have now been told to withdraw from the streets and return to their original posts in an attempt to stop the violence.
In the West Bank city of Ram Allah, Fatah has enforced a general strike, closing shops and private schools in a show of force.
The Hamas-led government has closed all ministries to protest against Fatah attacks on government buildings.
A Hamas official from the finance ministry has been released by his Fatah kidnappers, who told him his abduction was intended to send Hamas a message to end the violence, Hamas officials said.
Hamas has denied claims by Fatah gunmen that they had kidnapped a Hamas minister.
Ismail Haniya, the Palestinian prime minister, told his cabinet on Monday that the tension in Gaza had eased, but worried that the violence could spread to the West Bank.
"We reiterate to our people to be responsible, not to spread the circle of disagreements and conflict, and not to transfer events to other parts of the nation," he said.
Many shops in Gaza have been closed due to fears of renewed violence, and the agricultural ministry was torched early on Monday.
Abbas has been locked in an increasingly bitter struggle with Haniya's government since efforts to form a unity coalition stalled.
The government's problems in paying the salaries of its workers, many of them from Fatah, have increased tensions between the groups.
The Hamas-led government is short of funds partly because of an aid embargo imposed because of its refusal to recognise Israel and renounce violence.
Israel has also been withholding tax revenues and its effective blockade of Palestinian territories has ruined the local economy.
Eight people were killed on Sunday during clashes between Fatah and forces loyal to Hamas in the worst internal fighting for months.