Yes, there are many side effects of hair coloring dyes because they contain effective oxygen varieties that fight antioxidants in the hair and reduce the positive action which leads to major problems
In medicine, a side effect is an effect, whether therapeutic or adverse, that is secondary to the one intended. Although the term is predominantly employed to describe adverse effects, it can also apply to beneficial, but unintended, consequences of the use of a drug or a any other substance. That said, permanent hair colors contain ammonia and peroxide that neutralize the natural pigment in the hair and subsequently damage it. Hair dyes contain paraphenylenediamine which causes allergy like itching, skin irritation, redness or swelling. Conjunctivitis, asthma and cancer are other adverse events of hair dyeing that have been reported. The details could be found in the publication attached.