This misunderstanding is so pervasive, that some have even stated that airports might lose their federal funding unless they use coal tar. While I won't get into the politics of procurement, it is safe to say that not only do airports around the country use other materials besides coal tar sealants, but many have been looking for alternatives for a very long time.
The reason that coal tar sealants have been historically used at airports is their fuel resistance. If you remember your chemistry class taught you that "like dissolves like," then you'll understand that asphalt (a petroleum by-product) is damaged by fuel (a petroleum by-product) leaks from aircraft. Airport can't afford to have deteriorating pavement both for structural strength and also to minimize any debris on the field that might get sucked up in an engine.
However, there are problems with coal tar sealants and fuel resistance and it has nothing to do with environmental or human health issues. Several sources cite pavement damage when coal tar sealants crack and allow jet fuel to sit in the cracks and deteriorate the pavement. The first entitled "Fuel Resistant Sealers and Binders for HMA Airfield Pavements (2009)," is a study of fuel resistant surface treatments. It says that differential expansion and contraction of a coal tar sealants leads to cracking (p.1). The study, while not approved by the FAA, was funded by the FAA.
The second source is the Army Corps of Engineers that found this same cracking phenomenon. This report is: "Rejuvenators, Rejuvenator/Sealers, and Seal Coats for Airfield Pavements (2003)."
When the City of Austin passed the nation's first coal tar sealcoat ban, language was included that gives the airport an out if they couldn't find a suitable product to replace coal tar at their most critical location, the aircraft parking areas. This is where a lot of fuel drips from the planes and fuel resistance is critical. This is a VERY small percentage of the area sealed at the airport.
Since then the FAA has done research and has found products that can stand up to fuel drips. Several sealants were found to perform well.
Other airports around the country have used alternatives to coal tar sealants. The San Francisco Airport, the John Wayne Airport (Orange County, CA) successfully used a fuel resistant product as well. More on this particular product can be found at this link.
Since 2005 the Austin-Bergstrom International Airport has been using only asphalt-based products without any loss in FAA support.
So contrary to what you might hear, there are suitable alternatives to coal tar sealants being used today at airports around the country. All without any loss in funding or support from the FAA.