The school compound at Ain al Hilweh houses four UNRWA schools attended by more than 3,000 Palestinian children.
The compound was damaged in the battle for control of the camp earlier this month between armed Islamist groups and Fatah, which mostly controls the camp.
“This is a grave violation of the inviolability of UN premises under international law which compromises the neutrality of UNRWA installations and undermines the safety and security of our staff and Palestine refugees,” said Dorothee Klaus, the director of UNRWA affairs in Lebanon.
Ms Klaus called for the protection of all UNRWA facilities.
“Schools are meant to be a sanctuary for children and zones of peace where children learn and play. They must never be used for armed conflict, clashes, violence and fighting.”
The clashes, which began in late July and continued into August, were some of the most fierce fighting the camp has faced in years.
At least 2,000 refugees were displaced in the fighting, UNRWA said, while dozens were injured and at least 13 were killed.
In July, high-profile Fatah commander Abu Ashraf Al Armoushi and four of his bodyguards were assassinated in the violence.
For more than a decade, Ain Al Hilweh, notorious for harbouring criminals and Islamist militants, has been the site of a power struggle between a network of militant extremists and the Fatah movement’s security apparatuses.
Fatah, the strongest political faction in Lebanon’s Palestinian camps, has for years tried to contain the presence of outlaws and smaller networks of insurgents who seek to gain control of Ain Al Hilweh.
The overcrowded camp holds more than 50,000 registered refugees, many of them from coastal towns in northern Palestine, now northern Israel.
Lebanon’s Palestinian refugee camps are fertile ground for fugitives and outlaw groups to flourish owing to a contentious decades-old agreement that, for the most part, prevents Lebanon’s military from entering them.