I think the story has a good chance at being true.

The story has been told in multiple versions going back as far as November 1940. The story reportedly happened around Amsterdam. In Early 1940 The British Bomber Command was still dropping more leaflets than bombs. It is not inconceivable that during this time a bomber crew would have thought it funny to drop a wooden bomb on a fake airfield. Wooden practice bombs would likely have been available.

Pierre-Antoine Courouble actually researched this topic and wrote a book "The Riddle of the Wooden Bombs." In it he has an interview with A German Soldier who claims to have worked at a dummy airfield and to have had wooden bombs dropped on the airfield in late 1943 near Berlin. This would be a likely scenario as the US and Britain both dropped leaflets over Germany late in 1943. Again not inconceivable that a bomber crew would have thought it a clever joke to drop practice munitions on a fake airfield. This event is even more likely as the tale had been around for some time to inspire a crew to attempt such an event.

A joke similar to the one above was done in 1965 in Vietnam. CDR Clarence W. Stoddard, Jr. dropped a toilet on the Vietnamese forces to commemorate the 6 millionth pound of ordinance dropped. He dubbed the mission Operation Sani-Flush. The toilet was a broken one from the USS Midway.

How exactly do you build a fake airfield? Is an airfield not just a long runway of concrete and a few hangers? How can you fake them? Was it the planes that were faked?