Is it permissible to take a hairdryer from the school's inventory after it has been damaged by the school, or is that considered a sin?

It is not permissible to dispose of school property except in ways that serve a beneficial interest, for public funds are sacred. A severe warning has been issued against those who unlawfully take from it, as the Prophet (peace and blessings be upon him) said: "Indeed, some men delve into Allah's wealth without right, and for them is the Fire on the Day of Resurrection." An exception is made for items that would be thrown into the trash due to not being needed or having no value; there is no harm for someone to take them and benefit from them or give them to someone who can benefit from them.