Israel's military said about 100 people rushed the tomb of the biblical patriarch Joseph, which is located in the Palestinian city of Nablus. They were pushed back by Palestinian security forces who arrived on site, but not before setting parts of it aflame.
"We view this incident with gravity and strongly condemn any attack on holy sites. We will find and arrest those who set the fire," the military said in a statement.
The unrest that has engulfed Jerusalem and the occupied West Bank, the most serious in years, has claimed the lives of 32 Palestinians and seven Israelis.
The Palestinian dead include 10 knife-wielding assailants, police said, as well as children and protesters shot in violent demonstrations. The Israelis were killed on the street or buses in random attacks.
There was, however, a respite from the near-daily attacks on Thursday.
The U.N. Security Council will hold a special meeting to discuss the situation. No resolution is planned for Friday, but there might be an attempt to get the council to issue a statement aimed at urging the two sides to curb the violence.
Hamas, which controls the Gaza Strip, called for "rallies of anger and confrontations" on Friday in all West Bank cities.
The unrest has been triggered in part by Palestinians' anger over what they see as increased Jewish encroachment on Jerusalem's al-Aqsa mosque compound, which is also revered by Jews as the location of two destroyed biblical Jewish temples.
Israel says it is keeping the status quo at the holy compound.
U.S. Secretary of State John Kerry has said he plans to travel to the Middle East soon to try to calm the violence.
"A suggestion was raised that John Kerry and I and King Abdullah (of Jordan) and others would meet," Israeli Prime Minister Benjamin Netanyahu said on Thursday.
"I have no problem with that, we did that a year ago, it was actually fruitful. It could happen again," he said, adding that he was willing to meet Palestinian President Mahmoud Abbas.