Abayat was killed in a car explosion on Thursday
Four Palestinians, including a woman and a policeman, were reported killed and least 16 injured in exchanges of fire in the West Bank, while the Israeli army said a soldier was seriously wounded by a sniper.
Hospital sources in Gaza said that a Palestinian had died of gunshot wounds to the head following clashes at the Karni crossing between Israel and the Gaza Strip. A boy was earlier killed when an Israeli tank shell he picked up exploded.
The killings occurred as thousands of mourners joined the funeral processions of three Palestinians - including local militia leader Atef Abayat - who died when their car exploded near Bethlehem.
The Associated Press quoted Kamel Hamad, a colleague of Mr Abayat, as telling the crowd that Palestinians would continue their struggle "as long as there is one soldier or one settler in our land".
The BBC's Barbara Plett says the Israeli prime minister, Ariel Sharon, appears to be putting the Palestinian leader, Yasser Arafat, under increasing pressure - possibly with the aim of toppling him from power.
Israel sent tanks and bulldozers into Bethlehem and Beit Jala on Friday morning, dismissing appeals from Washington for restraint.
Tanks started rolling into the two West Bank towns after Palestinian gunmen attacked Gilo, a Jewish neighbourhood on occupied land, in response to the killing of Mr Abayat in a car explosion on Thursday.
Violence has been escalating in the region since Palestinian militants killed the Israeli Tourism Minister, Rehavam Zeevi, in a Jerusalem hotel two days ago.
US pressure
The Palestinian Authority has arrested several suspects in connection with the assassination, but rejected Israel's demand to hand them over.
In an increasing war of words, the Palestinians have accused the Israeli Government of returning to violence and providing more excuses for international terrorism.
The death of Mr Abayat, the Bethlehem military leader of Palestinian leader Yasser Arafat's Fatah faction, came just after Israel had promised retribution for Mr Zeevi's assassination.
Our correspondent says that while officially the army suggests he was killed by his own bomb, privately Israeli security sources say he had been targeted by Israel.
Israel has renewed its policy of tracking and killing militants on its most wanted list, saying it is forced to do so because Mr Arafat does not arrest them.
Call for restraint
The US State Department, which is attempting to hold together a coalition for its war on terrorism that contains many Middle East states hostile to Israel, had called on the Israelis to exercise restraint following the assassination of Mr Zeevi.
Mr Zeevi, a highly controversial hardliner and the first elected official to be killed by Palestinians since the creation of the Jewish state, was buried in an emotional ceremony on Thursday.
US spokesman Philip Reeker had said that an overreaction by Israel to Mr Zeevi's killing would let terrorists derail recent steps toward peace with the Palestinians.