Is it permissible for an employee to take a sum of money in exchange for signing a "mission travel" form for a trip that did not occur, knowing that his manager was the one who suggested it to increase his salary?

If the manager is authorized to reward employees, then there is no objection to it. However, if he does so on his own initiative, without permission, then it is not permissible for the employee to take the money; otherwise, he would be consuming public funds unlawfully and cooperating in sin and aggression. He must refuse it, based on the Almighty’s saying: "And do not cooperate in sin and aggression" (Quran 5:2), and the Prophet's (peace be upon him) saying: "Indeed, there are men who delve into Allah’s wealth without right, and for them is the Hellfire on the Day of Resurrection." To ascertain the legality of the reward, one should refer to the regulations and higher authorities.