The main reason why the flame has no shadow is because the flame itself is a source of light.
A shadow is the surface area which is less bright than its surroundings because something is blocking light partially or fully from that area. Therefore, a shadow is nothing but a darker area with the absence of light. If the light source behind the flame is brighter than the flame, then the flame would cast a shadow. However, it's usually not possible because the fire in itself is the brightest source of light and also because the flame is transparent to some extent.
Happy reading.
Thanks and Regard.
Follow @jakerv