What is the ruling on an employee signing in and out on his official day off—such as Friday—without working, and being paid for it, with the knowledge of his manager, for the purpose of improving his financial situation, given that company regulations allow this?

If what an employee does violates labor regulations, then it is not permissible for him to do it, and the wage he receives for the days he does not work is not lawful for him, even if the manager knows and remains silent. However, if it is permitted according to labor regulations and authorized by someone with the authority to permit it, then there is no blame on him. The basis for this is the obligation to fulfill contracts and conditions, as Allah Almighty says: "O you who have believed, fulfill [all] contracts," and as the Prophet, peace and blessings be upon him, said: "Muslims are bound by their conditions."