Is it illegal for the Inland Revenue Department to run errands part-time?

Legal analysis: not illegal, just a service. Just like express delivery.

If you invoice other companies to help them evade taxes, it is illegal to evade taxes.

Legal basis: Article 53 of the Constitution of People's Republic of China (PRC). People's Republic of China (PRC) citizens must abide by the Constitution and laws, keep state secrets, cherish public property, observe labor discipline, observe public order and respect social morality.