Blue Feathers
Despite the existence of blue-feathered birds, all birds lack blue pigmentation. When you see blue feathers it's an optical illusion. Birds that appear blue have a feather structure that reflects blue light. A genetic mutation could conceivably change the structure of a sparrow's feathers to allow this phenomenon.
Food
Food choices may also explain a blue sparrow. Food often affects feather color. For instance, a flamingo's pink hue results from the red shrimp and algae it consumes. The sparrow may have eaten something that did the same, although the bird would have needed to eat the same food for some time.
Other Theories
Some critics suggested that Shears had used a photo-editor to alter the bird's color. Examination of the photo showed no evidence of tampering or creative editing. A rather wild theory suggested the bird was the offspring of sparrow/budgerigar crossbreeding. Budgerigars, or budgies, are better known to Americans as pet store parakeets. Even if a budgie did mate with a sparrow the chance of fertile eggs is highly unlikely.
Dye
Chances are good that Shear's blue sparrow inadvertently came into contact with some substance that dyed its feathers possibly by bathing in water stained blue by human activity. This theory seems most likely, especially as people reported blue corellas in Sydney around the same time as Shear's sighting. Corellas are small members of the cockatoo family and are usually white.