"Use" needs a bit more explanation.
The image files can be located anywhere: internal drive, external drive, network drive.
The catalog file must be located on a local drive (internal or external drive). Since LR is designed as a single-user, single-computer system, it prohibits the catalog from being opened from a network drive so that multiple users do not try to use it at the same time.
You can install LR on any computer whether it is on a network or not (I would say that 99.999999% of all computers today are on some type of network).
The problem comes from the LR catalog file. That must be stored and opened from a drive connected directly to the computer. That file can not be store on or access from a network drive.
The images can be store on a networked drive whether another real computer, Server or a NAS system.
So your IT department is partially correct and partially incorrect. Whether they want to spend the money to have LR installed on all the computers your classes may use is a different story.
If the students are supplying their own computers then the school would need to flit the bill to install LR on all your students computers or it would be up to the students parents to supply that program.
Ill make a few assumptions.
1. You have a computer.
2. Each student has a computer
3. All of the computers are on a shared network.
As for teaching.
If you want to demonstrate the functions of Lightroom on a given set of images and have the students follow along and do the same I would have a catalog and a series of folders on your machine containing the images.
I would duplicate the catalog and series of folders to the student computers via the network. This can be done before each class so the students will always be starting at the same point as you, regardless of any additional experiments they make. You are the "Master" controller.
I wouldn't store any images on the network in this instance as 20 students accessing the same images is going to create unnecessary conflict and change warnings in everyones catalog.
If you want the students to have the ability to practice what they learn on their own images I recommend they each have a separate catalog for their own use.
The problem with school computer networks is usually that the users are not allowed to have write access to the C: drive on Windows (or the internal drive on a Mac). So the catalog file can't go there, the network administrator has no problem allowing the users to have write access to network drives and put their catalog files there, but of course LR doesn't allow that. If that's the case, I know of no solution other than to beg the network administrator to make an exception for LR (and good luck with that).