Does white whole wheat flour need to be refrigerated?
The only difference between regular whole wheat flour and white whole wheat flour is the variety of wheat from which it is ground.  Regular whole wheat flour is ground from red wheat while white whole wheat is from a lighter pigmented wheat and thus the lighter color.  In both cases the germ is still intact and thus as Jonathan Campbell stated, it will go rancid more quickly than white flour.

Keep it in a cool dark place and use up quickly if you can't refrigerate or freeze it.

When you keep it in the refrigerator or freezer and plan to use it for yeast breads, remember to allow it to warm to room temperature before using for best results with your yeast.