The main effect of using a circular polarizer filter while capturing a rainbow can be seen in the following animation:
To generate this gif animation a circular polarizer filter was used. The filter was rotated stepwise to show the appearance and disappearance of the rainbow. This animation of the rainbow at Niagara Falls contains a total of 13 single exposures.
A polarizer filter can help to increase the saturation of the rainbow in an image, but it also can eliminate the polarized sun reflection in the image. Therefore, depending on the used focal length it might be necessary to capture several images with different polarizing filter settings to bring out the colors of the rainbow in the entire image.
The use of a tripod is highly recommended, but it is not mandatory thanks to post processing. The shown animation was captured hand-held while rotating the circular polarizer step by step.