What is the ruling on performing tasks unrelated to one's primary job as a programming مسؤول [official/responsible person], such as paying for website services for free and then profiting from this amount from the company? Is this money lawful or unlawful?

It is not permissible to take the value of the free invoice except with the company's knowledge and consent, because it gives you the money assuming that you paid it from your own funds. This falls under the general prohibition of taking a Muslim's money unjustly, and of cheating and deceiving them.