At present private schools pay less tax because they are afforded business rate relief. This is presumably in recognition that they are education some children who would otherwise fall burden on the State system. It is this relief that the Labour Party is considering abolishing.
Of course it's a philosophical point but I would contend that relieving somebody of a particular tax is not using anybody else's tax money. I also find it odd that not taking money from somebody is somehow seen as providing them with funds.
However, leaving that aside, some 600,000 pupils attend around 2,500 private schools. They are afforded tax relief of about £100m, amounting to about £160 per pupil. For this modest sum it is estimated that they relieve the State sector of about £2bn in costs. (This itself may be an underestimate because that amounts to about £3.3k per pupil and most analyses suggest a figure almost twice that is more accurate). Any other facility which saved the government twenty times what it has to provide in tax relief should be encouraged, not threatened.
There is the usual element of envy with this proposal as there always is with private healthcare and private education. Whilst it is true that many wealthy people send their children to private schools (and would continue to do so even if the fees were raised to cope with the lost tax relief) private schools also educate pupils from far less affluent backgrounds. Many of them make huge sacrifices to afford school fees, often forgoing the holidays and new cars more readily afforded by parents whose children attend State schools. It is they who would be hit by this proposal and instead of threatening the excellent facilities provided by private schools politicians would be better off devising ways to bring the often appalling State system up to the same standards. And this is not a matter of cost as most studies show that the costs for State education and private education (discounting boarding costs) are comparable. But much of the cash spent on State education does not get near a classroom but is swallowed up in adminsitration costs mainly in the form of interfering Local Education Authorities.