They have accused the UN, which started up the educational scheme in 2007, of corrupting young people.
Hamas, who have control over the Gaza strip, run alternative camps involving military-style marching and Islamic instruction.
John Ging, the local UN chief, described the attack as “cowardly and despicable" and said it was another sign of the urgent need to tackle rising extremism in Gaza.
He promised the site would be rebuilt.
The attack follows Israel’s decision to substantially ease the blockade of Gaza to allow more civilian goods in.
The UN Relief and Works Agency provides services for two thirds of the population of the area.
