TL:DR – No, rainbows do not cast a shadow.
A rainbow is an optical phenomenon. They are seen when light is refracted and reflected by water droplets in the atmosphere. They do not exist as physical objects in the same way that a solid object would and so do not block the path of light from the sun or any other light source so do not cast a shadow. You only see a rainbow when the light source is behind you and you are looking at the region in the atmosphere where the water droplets are present.
