All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
If the toilet seat is not impure, it is not obligatory to wash it after each use; even if it is impure, it is not obligatory to wash it as well.
Rather, it is only necessary to wash the clothes that a person wears to perform the prayer or Tawaaf, or to use them for what is only used when it is pure.
However, washing the toilet seat is part of cleanliness and Islam urges cleanliness.
Allah Knows best.