Does hair dye affect the kidneys?

The dyes used for internal x-ray imaging of flow in veins, etc, are chosen to be "radio-opaque", that is, opaque to x-rays. There are several materials available, and iodine compounds are one of them.

They are chosen to be benign to the body, and non-reactive. For obvious reasons. It could be seen in the kidneys, or arteries, or veins as required, on an x-ray. It will be expelled as an unwanted material in the usual way.