What chemicals in hair dye can have consequences for your health?

PPDE. Most common cause of allergic reaction or contact dermatitis to hair dye is ppde (para-phenylenediamine). This can cause mild to severe reactions ranging from redness, swelling and itching of the scalp, eyes, and skin around the hairline to severe anaphylactic reaction. Hairdressers who are allergic can also develop rash of their hands. Only cure is to avoid this chemical.
All of them. Maybe. No one knows long-term health risks of most of the chemicals in our environment, including those we intentionally put on our scalps. That being said, hair color dyes have been around for decades, and used by women for their entire lives. Right up until they die of something that (probably) is totally unrelated to hair dye chemicals. So, weigh the (low) risk/benefit (?) and decide if it's worth it!