Does money earned from online teaching become affected by using a copyrighted, pirated book, and is there any doubt in using a copyrighted book that has been officially purchased for online teaching?

If your brother bought the book and used it in lessons, or if it was published on public websites with the permission of its owners, then there is no sin upon him. As for books and references whose owners do not permit copying or downloading, he is not allowed to infringe upon their rights. However, this does not forbid his earnings from the lessons.