Clashes have broken out overnight in several areas of Lebanon following the funeral of the senior intelligence official, Wissam al-Hassan.
The most serious confrontations were in the northern city of Tripoli, where at least three people were killed.
In Beirut, gunmen exchanged fire and police used tear gas to disperse crowds outside the prime minister's office.
Gen Hassan and at least two other people were killed in a car bomb attack in the capital on Friday.
The head of the intelligence branch of the Internal Security Forces had been an outspoken critic of the government of Bashar al-Assad in neighbouring Syria and its allies in Lebanon, and Lebanese opposition figures have blamed Damascus for the attack.
They have called on Prime Minister Najib Mikati to resign. He offered to do so on Saturday but President Michel Suleiman asked him to stay on in the national interest.
Thousands of people attended Gen Hassan's state funeral in Beirut on Sunday, which rapidly became a political rally against both Mr Mikati and Syria.
Addressing the funeral, former Prime Minister Fouad al-Siniora, a prominent member of the Western-backed 14 March opposition alliance, blamed the government for his death, and said it must stand down, insisting: "No dialogue over the blood of our martyrs."
A group of angry protesters broke away from the funeral, held at the Rafik Hariri mosque, and attempted to storm the prime minister's office. Police fired into the air and used tear gas to break up the crowds.
Later, in southern districts of the capital, security forces and gunmen exchanged fire.
More serious clashes were reported in Tripoli on Sunday evening, where a nine-year-old girl was among three people killed. Reports say the girl was hit by sniper fire.
Mr Siniora subsequently called for calm and condemned the violence as no way to try to replace a government.
Dozens of people have set up camp outside Mr Mikati's office building, calling for Mr Mikati's cabinet - which is dominated by the pro-Syrian Shia Islamist movement Hezbollah and its allies - to stand down.
One of the protesters, Ahmad Jardali, said they were peaceful but wanted the government to go "in order not to have more assassinations and to have security in the country".
The US has said it will help the Lebanese government with its investigation into the Beirut bombing.
Gen Hassan, 47, was close to the 14 March alliance and the family of its leader, former Prime Minister Saad Hariri.
The intelligence chief led an investigation into the 2005 assassination of Mr Hariri's father, Rafik - for which four Hezbollah members have been indicted by a UN-led tribunal - and had recently organised the arrest of Michel Samaha, a former minister accused of planning a Syrian-sponsored bombing campaign in Lebanon.
Syrian troops withdrew from Lebanon in the wake of Rafik Hariri's murder, ending a 29-year military presence.
The BBC's Wyre Davies reports that there are concerns in Beirut that Damascus is able to reach into Lebanese society both directly and through its allies.