What is the ruling on tying hair during the performance of Hajj or Umrah rituals, and does anything become incumbent upon the one who does so?

The prohibitions related to the head for someone in the state of ihram for Hajj or 'Umrah are: removing hair from the head, perfuming the hair of the head, and covering the head for men. Tying the hair with a string or similar is not among the prohibitions, because it is not considered removal, perfuming, or covering.