Ramzan,
Your wife does not need to pay for the tyre. When you take driving lessons, you need to have insurance cover in place. If you learn in your own car, then you need to
arrange for learner's insurance privately. However when you learn with a driving instructor, the latter is responsible for making sure the car is properly insured.
Any incidents/accidents that happen during driving lessons can be claimed from that insurance. Your wife's instructor does not want to claim from his insurance because he will lose his no claims bonus. He also does not want to pay from his own pocket, so he's demanding the money from your wife. It's a cheap trick! Don't pay up and tell him that all driving instrutors need to have insurance cover when they are teaching people to drive as learners are more prone to mistakes and incidents.