Foreign pro-Palestinian activists have been camping outside the house in recent weeks in protest of the court ruling. Seven were detained for questioning on Sunday, police sources said.
International leaders, including Condoleezza Rice, the US secretary of state, have spoken out against plans to expel the family.
Symbol of resistance
The al-Kurd's fight to remain in their home has become symbolic of Palestinian resistance to the expansion of illegal Israeli settlements in occupied east Jerusalem overall.
The family were given the house back in 1956 by UNRWA - the United Nations agency for Palestinian refugees - after they, like many others, fled Israel to east Jerusalem following the creation of the Israeli state in 1948.
Another 27 Palestinian families who evacuated to east Jerusalem in the 1940s, which was then under Jordanian control, were also allocated housing in the Sheikh Jarrah neighbourhood by the UNRWA.
However, after Israeli forces captured and annexed east Jerusalem during the 1967 Six Day war, Israeli settlers registered their claim to the land. The al-Kurd's say that the 19th-century Ottoman document upon which the title deeds are based is a fake.
The move comes just two days before municipal elections. In Jerusalem, two leading candidates for mayor, Meir Porush and Nir Barkat, have both spoke out in favour of Jewish settlements in the east of the city.
Hatem Abdelkader, a spokesman for Salam Fayyad, the Palestinian prime minister, reacted angrily to the eviction.
"They want to expel Palestinian residents from Sheikh Jarrah. It is an escalation before the municipal elections," he said.
The eviction happened on the same day as the international quartert for the Middle East - the US, Russia, the European Union and the United Nations - met in the Egyptian Red Sea resort of Sharm el-Sheikh.
Leaders urged the Israelis to stop the construction of Jewish settlements on occupied Palestinian land.