Yeah, Kiwi nailed this...There are many, many, many things wrong here, and not just wrong, but potentially fatal to the birds. First of all, if the birds are inside of a cage where they are chewing the paint/powdercoating/metal off of the bars, this is a very dangerous situation that is going to end in tragedy, as it usually does. Any time this happens the birds must be moved into a new cage, preferably one that is not painted or coated with anything that can be chewed off by the birds.
However, there is a very good reason that these birds are chewing on the cage!!! It's because they are absolutely bored to tears! These guys have the intelligence of a 4 year-old human child, and they have one perch and one toy? That's it? They should have a minimum of 6-10 different toys at any one time, and each toy should be a different type of toy-one for chewing (wooden), one for shredding, one for foraging, etc. And for both their mental and physical health they need multiple perches, each made of different materials and different sizes; if they are simply sitting on one perch (probably a wooden or plastic dowel) and that's it, they are going to get Bumblefoot and other foot conditions. The main issue though is the lack of stimulation! They never get out of the cage to fly, so no exercise or interaction/stimulation there, and they have 1 measly toy. They are becoming more and more aggressive and most likely starting to over-preen due to boredom and stress. This is going to turn into self-mutilation and feather-destructive behavior very quickly.
Now, the breeding situation...This is awful, and for a bunch of reasons...First of all, do they have both a Cuttlebone and a Mineral Block inside of the cage at all times? If not, the female is going to eventually develop a calcium-deficiency and is at great-risk for becoming Egg-Bound, which is 100% fatal if not intervened with instantly.
The fact that the female is laying clutch after clutch after clutch is worrisome. Cockatiels are prone to chronic egg-laying, and removing their eggs is causing her to lay more and more. So they either need to remove each real egg as they are laid and then freeze them overnight or boil them and then put them back in the nest, and do this with each egg as she lays them, or they need to buy fake Cockatiel eggs and simply replace each egg with a fake one. And you need to leave the eggs in with the female until she loses interest in laying on them on her own. At that point she'll just stop laying on them and she'll start ignoring them, then you can throw them out. But if they are removed before she loses interest in them on her own, she's just being triggered to lay another clutch.
These birds shouldn't be breeding in this situation anyway...I don't think anyone has yet to ask what the bird's regular, daily diet is, but I'm going to assume it's a bagged seed-mix? Is that it? Or are they eating a pellet diet? Do they get fresh veggies every day? How about some Egg-Food for the female? Laying clutch after clutch after clutch of eggs is extremely stressful on the female's body, and it quickly will become apparent that she's starting to suffer physically because of it; their feathers eventually start becoming dull and patchy (and they also often start plucking), they become very skinny, their eyes become dull, they often start open-mouth breathing all the time, it actually looks like they are panting like a dog does constantly, 24/7, their bones become brittle and they start to have trouble moving/walking without pain, and this is because the calcium is being leeched from their bones to make eggs, etc. This is what is going to happen if these birds continue to lay clutch after clutch.
Is there a nestbox in the cage with them? If so, then you need to talk to whomever is "in-charge" or whomever owns these birds about trying to stop them from breeding anymore, and the first thing they need to do (after the cuttlebone, mineral block, Egg-Food, etc) is to remove any nest-boxes or nesting material that is anywhere in the cage to try to discourage them from breeding. The next thing is to try to get a handle on their natural light schedule. They should be on a natural light schedule, where they are able to see the sunrise and sunset through a window naturally, and they should ideally be covered and put to bed right as they watch the sunset, and be awakened and uncovered with the sunrise. This will help tremendously to calm their hormones.
I commend you for trying to help these birds, it's not your responsibility and I thank you for caring about them, as this is a bad situation for parrots to be in. Unfortunately parrots do not make good classroom pets/animals at all, simply due to their needs and how much stress negatively effects them...and also because it's not a situation where they are going to be able to get any out-of-cage-time at all, they are locked inside a cage 24/7, and if this is the case and they don't have a ton of different toys and foraging activities inside of the cage they are locked in 24/7, that's when they become stressed, they become aggressive, and they start breeding continually, which starts the serious health issues for the female. They just are not good classroom animals. No one is there after what, 4:00-5:00 at the latest? So they get no stimulation, no time to fly around, no exercise at all, they just sit inside a cage with literally nothing to do. 1 toy inside any parrot's cage is shameful, let alone with 2 cockatiels.
It doesn't sound to me like whomever is in-charge of this vet-lab program knows anything at all about caring for parrots, their needs, their wants, their requirements, and how horrible allowing them to constantly breed is...And if they removed the eggs anyway, that indicates that they aren't purposely trying to breed them anyway, or wanting them to breed, right? So why are they putting them into a situation where they have nothing to do but breed? Don't get me wrong, I'm thrilled that this isn't a situation where they're purposely trying to get them to breed so that they can raise baby birds....that would be absolutely horrible. But if an adult is going to take-on keeping a bonded breeding-pair of parrots in their classroom at a school, then they should know enough to do all of the research and educate themselves, which this person obviously hasn't done...They would be much better-off either having a single bird, or having 2 males or 2 females, or putting these two birds into separate cages right beside each other so that they can no longer mate, but they can still see each other, talk to each other, and interact. Sometimes this is what has to be done for the welfare of the female.
I hope that you're able to talk to this teacher or whoever it is that runs this program, sit down with them and explain to them that these 2 birds are becoming aggressive because they are both extremely stressed due to#1) Utter and complete boredom, #2) A total lack of mental stimulation, #3) A total lack of physical exercise, #4) Hormonal issues caused by the constant breeding and the situation with the constant egg-laying, etc. Tell them that the female is in great-danger of developing serious and potentially lethal health conditions due to the chronic egg-laying, and that she does not have any of the things that she needs to keep her from becoming calcium-deficient, such as the Cuttlebone, a large Mineral Block, Egg-Food in her food bowl every day, fresh veggies every day (dark, leafy greens specifically), etc. And someone, I don't know who but someone is going to have to bite the bullet and go out and buy a lot of different types of toys for these guys, they must have at least 6-10 different toys, the more the better, along with a bunch of new perches made of different materials like different branch-perches of different diameters, at least one sand-perch to keep their toenails from needing clipped, etc. They need some ladders to climb. They need some rope toys or plastic chains hanging that they can climb up and down and swing from. They're extremely intelligent and they are being slowly killed due to a total lack of mental and physical stimulation. If all these 2 birds do is sit inside a locked cage with no toys and nothing at all to do all day long for years but breed and chronically lay eggs, then they are much better being re-homed to either a private individual who has parrot experience and that does not want to use them as breeders, or at the very least they need to be surrendered to a reputable Avian Rescue who can get them into a proper environment with proper mental and physical stimulation, get them on a proper diet, stop the breeding, and find them a loving and responsible forever home. Because this is not at all good for them...