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Thanks for this question. If your doctor did an exam after your got the hair dye in your eye and it was fine then it is nothing to worry about. It is possible the hair dye irritated the cornea and other tissues in the eyelids which is causing the burning sensation at the moment. Since the blurring of vision has improved i believe these other symptoms should pass as well. In the meantime you just need to monitor the symptoms and see if you develop anything new like redness, blurring of vision, pain in the eye or swelling, if those appear then you need to be re=-evaluated for corneal injuries.
If your doctor has not seen you after the hair dye incident, then it would be wise to see him IF the burning persists another 24-48 hours, but i think it should fade away as you dont have any worsening of the symptoms. Hydrogen peroxide in this amount shouldnt cause lasting damage to the eye as you must have washed it out at the time as well.
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