Is the amount of money paid to the accountant by the auditing firm, in exchange for him performing some tasks that the firm previously carried out, considered unlawful (haram) or a type of forbidden gift?

It is not permissible to accept the amount from the company that reviews the work except with the permission and knowledge of your employer. This is because this commission is due to your work, and it falls under the category of gifts to employees that are prohibited unless permitted. If your employer agrees that the company gives you a commission, then there is no harm upon you at that time.