What could cause the whites of your eyes to turn a brownish yellow colour?

Pigment changes. There are several causes of color changes in the white of the eyes. Most commonly, there is a slow yellowing that can occur with time. As well, increase in the size of small blood vessels can dull the white appearance. Rarely, jaundice and other diseases can affect the white of the eye. Any change in the eyes' color should be evaluated by an eye doctor.
Many things. Usually one thinks of jaundice, caused by a back up of pigment in the blood called billirubin. This is usually associated with problems or diseases of the liver. Iron overload syndrome could potentially cause this as well. I am sure that there are many other causes. If you have a problem, please see your pcp for an evaluation and discussion of a potentially severe problem.