This may be the article you are talking about:
https://archive.md/zht5Z
I am not sure what you are referring to by 'net zero tax'.
The UK tax on flying is Air Passenger Duty, which depends on the destination country as well as the cabin of travel. APD is the same for all UK airports, except that Scottish Highland and Northern Ireland airports have some exemptions. APD is levied on airlines and does not necessarily need to be passed on to the passenger, as seen when low-cost airlines have (in the past at least) charged less for an entire ticket than the APD due, or in many BA redemption tickets where they absorb the tax and charge you a "fee".
Here BA and Virgin are complaining about the charges that are levied by Heathrow privately. Fees that are meant to recoup the cost of security screening are directly passed on to passengers as you see on your ticket breakdown, while landing/parking charges are levied on airlines and just bundled up into the total fare charged.
While these charges are regulated by the Civil Aviation Authority, it is in the interests of Heathrow and its shareholders to extract maximum profit from customers by setting them as high as they can. The airport currently has no incentive to reduce costs, particularly because of its monopolistic and premium position, and at present this is allowed by the legal/regulatory framework.
The airlines contend that they, and/or other organisations, should be allowed to operate parts of the airport or at least have more say in how it is run.
If any cost savings found lead to Heathrow dropping its charges, one can expect airlines to raise their fares by the same amount.