Because fire is a source of light and mostly transparent, it doesn't block light like a solid object. If you hold a flame in sunlight, you might see a "ghostly" shimmer on the wall, but that's just hot air moving, not the fire itself!
Shadows are formed when an opaque object blocks light. However, if you try to find the shadow of a candle flame or a campfire under bright sunlight, you will notice something strange: the flame itself isn't there.
Fire is a plasma-like state of hot gases undergoing a chemical reaction. Because it is a gas, it is mostly transparent.
If you look closely at the wall behind a flame in the sun, you will see a faint, flickering shimmer. This is not the shadow of the fire.
A flame can only cast a shadow if the light source behind it is much brighter than the flame itself AND the flame contains a lot of soot (unburned carbon). A very smoky fire might show a faint shadow because the solid smoke particles are finally blocking the light.
Fire is an "invisible" guest at its own shadow party. What we perceive as its shadow is actually just the air dancing because it's too hot to stay still.