Yes, because the blood flowing through your feet will also get colder as it loses heat to the surrounding tissue. When it circulates back into the upper body it will be warmed up again thus reducing core body temperature. There are a couple of limitations;
1 there's nothing to stop your body burning fats to maintain your core temperature despite the fact that your feet are still freezing!
2 As your feet get colder the body tends to restrict the flow of blood to the skin thus slowing the rate at which heat is lost to the exterior.