The rabid beagle's tag reads Canis familiaris, an obsolete classification. Tame dogs (familiaris) are now considered to be a subspecies of the wolf Canis lupus (which Fox correctly identified on the mountain), rather than their own species. Since news of species revisions travels slowly, it is realistic that not everyone has heard of the change.
Early in the movie, Mr. Fox sits at the breakfast table with a newspaper. In it, he notices a picture of a tree for sale, which has landscape (horizontal) orientation. When he cuts the picture, he does so as if it was a portrait (vertical orientation). In the next scene, he holds the clipping up, and it is in landscape orientation again.
When Mr. Fox looks out the tree's window and notices the distant Boggis, Bunce and Bean farms for the first time, they are lined up in that order left-to-right on hilltops. But when he later shows a map of the farms to Kiley, the Boggis farm is clearly situated between Bean and Bunce.
Frank Bean, inside a caravan, on the phone says "They took everything, let me call you back Petey." After a quick burst of anger he exits the caravan, and Petey is sitting right outside fixing a bike, without a phone (not that you'd talk to someone on the phone who is right outside anyway).
In long-shots, there is a building between the second and third of the four silos on the Bean farm ("E" and "A"). In closer angles, the silos are equally spaced and the building is not readily visible.
Bean's helicopter is clearly modeled off of the helicopter in Magnum, P.I. and has the exact same paint scheme when viewed in long shots. However, a close up shot of the pilot in the cockpit shows the frame of the front canopy to be painted yellow, whereas it should be red in that section.
The film is clearly set in England, according to Mr. Fox's morning paper and other clues. But the wolf's alpine backdrop is like something out of the Rocky Mountains, and Kiley is an American opossum--there are no opossums or any marsupial relatives native to Europe. On the wackbat-trophy's champion-list there is an "M.K. Silvery-Marmoset", a creature found only in very limited areas in Brazil.
Mr. Fox says the Latin names of the animals in the film. When he reaches Mouse, he incorrectly calls Mouse Microtus Pennsylvanicus. Microtus Pennsylvanicus is actually a Meadow Vole. Mouse's Latin name should've been Mus.
When Latin names of the animals are displayed in the film they are not written in the correct format. "Latin" names, more correctly termed scientific names, are standardized within the scientific community worldwide and written in a specific format, e.g. Genus species with the genus capitalized, the species written in lowercase, and both written either in italics or simply underlined. The film shows both genus and species capitalized.