What are the rules for racing on airport runways?
The adverse effect is once the layer of rubber gets too thick it starts to degrade braking performance, especially wet braking because the rubber is filling in the surface irregularities that help channel water out from under the tire.  Aircraft wheels will hydroplane at much lower speeds and with less water depth.  Airports remove excess rubber from touchdown zones once it gets beyond a specified thickness.

I would expect that the rubber from spinning car tires would be no worse than the rubber that builds up on touchdown zones on heavily used runways.  If an airport is closing a runway to use for drag racing, it can't be a very busy airport, but in theory a really heavy rubber build up could be a concern if infrequent high performance aircraft like business jets use it, you know, when no drag racing is going on.  That would be up to the airport operator what to do about it.