In this meaning, furred is not about animal hair. It is what happens when a surface slowly gets a soft, unwanted crust. A kettle furs up with limescale from hard water. Pipes can fur. A tongue looks furred when you are ill — a pale coating. Doctors used to talk about furred arteries, a fatty lining inside. British English uses this a lot for kettles; you also hear furred up. The picture is “as if a thin fur had grown on it,” which is how the meaning grew from fur, but the feeling now is dirt or deposit, not a nice coat. Though actually, the same spelling still means “covered in fur” in other sentences (a furred hood). Here we are only talking about that unwanted layer.