If they are searching for weapons or some other contraband, they need a warrant.
Landlords can perform a reasonable inspection of apartments, but they can't inspect drawers or personal belongings. I don't see why this school setting should be different.
Constitutional rights don't always trump property rights. For example, you have a Second Amendment right to keep and bear arms. However, a landlord could put a provision in a lease that says absolutely no firearms are permitted on his premises. If you sign that lease, you have effectively agreed to compromise your Second Amendment rights. You still have the right. You have simply agreed that you won't exercise it while you are on the landlord's premises.
Then take it one step further. Assume the landlord says in the lease that he reserves the right to come onto the premises at any time to inspect to make sure no firearms are present. If you sign the lease, you have agreed to the inspection. What is a reasonable inspection under those circumstances? Not too hard to conceal a handgun or a box of ammo in your underwear drawer, is it? Think a court would say it was unreasonable for the landlord to include that drawer in his inspection?
I doubt that there is a court in the land that would find these contractual provisions unreasonable or a violation of anyone's constitutional rights. If you think the landlord's rules are unacceptable, you have the right not to sign the lease.
For a real world example of this, look at the law dealing with picketing and demonstrating (otherwise known as exercising First Amendment rights) on private property such as shopping centers, where such activity is prohibited. ACLU and Union types absolutely HATE these restrictions, but they are generally upheld by the courts as legitimate exercises of property rights.
One exception to all of this might be if the landlord is a government entity. The original poster did not say if the school was a private or a state institution. A court might take a harder look at state action than with action by a private party, but in the circumstances we are talking about here, I doubt the result would be any different.