This has been bugging me for a while, and I need to figure it out because I want to use it in a fictional work: I remember reading (maybe in a National Geographic) about some airport built in the mid 20th century, where some farmers refused to sell their land, and so remained farming, between runways or at least very close to the runways. I had thought it was Heathrow, but I can't find any references on Google. Maybe somewhere in Asia? Any help is much appreciated!
You may be thinking of Narita airport - the main international airport serving the Tokyo metropolitan area. Many farmers had to be displaced to build the airport (not always ethically) and there were many violent clashes between the local farmers and the authorities in the 1960s and 1970s , and controversy and protests continuing into the 1980s and 1990s.
There are a small number of holdouts who still farm right next to the airport, and until recently those farms were blocking the building of a second runway. The second runway eventually did go ahead after agreements were reached with some of the farmers, but it was built shorter than originally planned.
A 1975 article (in a socialist paper) about the struggles of the farmers and about the protests
the Toho district, where farmers farm at the end of the runway
1999 article talking about one of the holdouts - and his farm right in the middle of Narita airport
Washington Post article talking about the holdouts
Edit: I hope this answer is sufficiently detailed for the sub. I am actually packing to fly from Narita tomorrow - after cooking a full Christmas dinner and hosting guests today - so I am almost delirious with tiredness.
I don't know of any large airports with farming but I do know that Huntsville, AL airport (HSV) had cotton farms between their parallel runways during the 1970's and it appears that they still do. http://www.airportimprovement.com/article/commitment-green-non-aeronautical-revenues