Militants loyal to Mr Abbas's Fatah faction set fire to the ground floor of the parliament building in Ramallah where the flames quickly leapt through the rest of the complex.
Gunmen from the same Al-Aqsa Martyrs Brigades also torched the cabinet building in Ramallah after hundreds of security officers, staffed largely by Fatah followers, forced their way into the premises and opened fire.
Prime Minister Ismail Haniyeh’s Ramallah office was empty when the Al-Aqsa gunmen entered.
Mr Haniyeh is based in Gaza, the militant group's stronghold, and does not have access to the city of Ramallah in the occupied West
Bank because of Israeli curbs on travel by Hamas members.
Black smoke spewed into the night sky out of the fourth floor of the five-storey cabinet building as fire engines rushed to the scene but were prevented by militants from trying to extinguish the flames.
Protesting against deadly violence between Fatah faithful and loyalists of the ruling Hamas movement, which controls the government, Fatah gunmen and security officers smashed windows, destroyed computers and tore up files.
In a bid to contain the situation, Mr Abbas issued an edict placing the Palestinian security forces on a maximum state of alert.
Outside the burning cabinet offices, a masked Al-Aqsa leader who refused to give his name called the protests a "normal response to Hamas aggressions against Palestinian Authority institutions and security service premises".
"We demand president Abbas sacks the government and appoints an emergency cabinet. We blame Hamas for this rift which risks provoking civil war," he told reporters.
Violence in Rafah
In another sign of the anarchy blighting the Palestinian territories, two people were killed and at least 17 wounded in Rafah, in southern Gaza, where Hamas gunmen tried to storm the preventive security service headquarters.
Hundreds of security reinforcements were dispatched to Rafah after Hamas militants managed to smash their way through a wall surrounding the headquarters by firing rocket-propelled grenades.
Shortly afterwards, a Hamas official was briefly kidnapped and then released in the neighbouring city of Khan Yunis by gunmen.
The clashes at the compound took place shortly after a Hamas supporter was killed in fighting that flared after a funeral for another Hamas follower who died from wounds suffered during a previous bout of infighting 10 days ago.
The three-storey building in central Rafah was riddled with bullets and many of the windows had been shot to pieces.
Cars parked within the compound had also been shot up by the Hamas followers, their windscreens shattered and tyres slashed.
Yussef Siam, head of preventive security in Rafah, said around 20 of his men were in the compound when more than 100 Hamas followers launched their attack.
In the gunbattle that ensued, a 34-year-old deaf civilian, Suleiman Zanoun, was killed after being caught up in the crossfire.
Although the Hamas gunmen were not wearing the uniform of a recently formed paramilitary force, Siam said they were members of the unit.
"This is a force that acts beyond the realms of the law. If they start targeting the government institutions, this will be the end of the Palestinian Authority," he said amid a mass of broken glass, rubble and bloodstains.
Witnesses said the clashes only broke up after hundreds of civilians massed at the scene and started chanting "Unity, Unity" as a sign of their disgust at the bloodshed.
Minister quits government
Meanwhile Palestinian Tourism Minister Judeh Murqos announced his resignation from the Hamas-led government in a statement to AFP, following the increased inter-Palestinian violence.
In a statement, Judeh said: "I confirm to you that I have resigned from the government due to the violence that has occurred in the Gaza Strip. He added: "I shall give other reasons tomorrow."
Mr Murqos, who comes from Bethlehem, was the only Christian member of the
government set up by the Hamas Islamist movement after its victory in January
elections.
He is also the first member of prime minister Ismail Haniya's government to resign.
At the core of the current tension is a referendum Abbas has called for July 26 on a manifesto for Palestinian statehood that implicitly recognises Israel. Hamas, which seeks to destroy Israel, has labelled the referendum a coup attempt.