A fever is a response by the immune system to help the body fight an infection. Raising the body temperature can cause bacteria and viruses to not be able to replicate as quickly and allow the immune system to gain an advantage in the fight.
Flushing during a fever is part of the skin's role in the thermoregulation of the body. Blood carries a lot of things to all parts of the body, including oxygen, nutrients, wastes, and in the case of flushing, heat. Blood vessels dilate and constrict in response to a raising or lowering of body temperature. When the body temperature decreases below our 'set point' of 98.6, capillaries in our extremities will constrict so that more blood, and thus heat, stays closer to our chest and abdomen. This reduces the amount of heat lost and helps to maintain a consistent temperature next to the most vital parts of our bodies.
When body temperature increases, as during a fever, blood vessels in our extremities including our face dilate. This allows more blood flow close to the skin, and extra heat can escape helping to reduce body temperature and maintain homeostasis. Even though a fever may be happening, this flushing can help keep the fever from reaching very dangerous levels.