This may be one of the most famous fashion myths of all time. An extension of this rule states that you shouldn't wear white before Memorial Day either. But why limit your color spectrum just because it's cold outside? You'll often find coats, sweaters and pants in "winter white," which is perfectly acceptable to wear.
Like most of these myths, this is another one whose relevance has faded over time. The reason it still rings true in some people's minds is that we tend to associate lightweight fabrics with light colors, and while the color white is perfectly acceptable in the winter, light fabrics, like linen, are not weather appropriate in many places from Labor Day to Memorial Day.