In the name of Allah, the most Beneficent, the most Merciful.
It is mentioned in Sunan Baihaqi that the Prophet of Allah Sallallahu Alahi Wasalam has ordered to bury the hairs and nails. The hairs should be disposed of in a respectful way, as it is a part of a human being. (Tuhfatul Ahwazi P.40 V.8)
In regards to your question the hair should be either buried or disposed of in a respectful manner. One will be allowed to dispose of the hair by putting it in a bin as the rubbish eventually is burnt. However, one should not throw it in the toilet or let it sink through the bath plugholes, as there will be impurities there.
Only Allah Knows Best
Mohammed Tosir Miah
Darul Ifta Birmingham