Of course, Heathrow could help to improve matters by choosing to ban some or all of cars/diesel cars/ diesel buses/taxis
I have argued this before re: Heathrow with Lord Adonis, and I've also argued in Jet Blast about the need to encourage more transport that takes the air pollution away all together (walking / cycling) or at least offsets it well out of the city (GW line will be electrified by then / electric buses or buses with flywheels etc.
The simple fact is that if there's going to be any chance of a 3rd runway at LHR, it has to come with some extremely tight air quality regulations. The airport cannot dictate to people that drive past it what sort of vehicles they use, but it can provide money, essentially through the largest Section 106 (planning bribe) agreement in history, to improve local transport.
The key to this is a good working relationship with TfL. Sadly, through being at the beck and call of Boris Jonhson, they remain wedded to the Fantasy Island project, and have recently sent reps round on an humiliating national tour to see this scheme.
The means are there to make it happen, and central London is already leading the way in terms of emissions zones and providing the cycle infrastructure. This is the only way a 3rd runway can happen, and it is going to take a serious rethink about how London's suburbs are ordered.