Burning Bush is also known as "Wahoo", a Native American word for arrow wood. Technically, the fruits are the hard, purple-brown appendages that formerly covered the bright red "berries". The "berries" are actually arils, a condition where the outer coat of the seed is bright and fleshy, serving the same basic function that the fruit normally would (i.e. dispersal).

Another good example of arils is the fruit of Magnolias. Here, too, the bright red objects are seeds with fleshy outer coats. Whenever you see something small, fleshy and red (like in the Magnolia and Burning Bush) you should suspect the dispersal strategy is for ingestion and transport by birds.