It is obvious that he has a delayed hypersensitivity
to hair dyes (Type IV hypersensitivity) that takes a few hours/days to manifest and lasts for several weeks sometimes.
He must take oral corticosteroids
and can also apply topical steroid
creams plus antihistamine tablets to control the current situation.
MUST be done after a few weeks to know which chemical is causing this reactin as the same chemical can be present in several other substances like toothpastes, washing powder, mouthwashes etc .