Is it permissible for an employee to use a company car for personal purposes, with the company bearing the fuel costs, based on the approval of his manager?

The ruling on using a company car for personal purposes varies according to the company's regulations. If the regulations permit it, then there is no objection. Otherwise, it is not permissible, for the Prophet, peace and blessings be upon him, said: "Muslims are bound by their conditions." If your manager owns the company or is authorized to give permission, then his permission makes it permissible for you to use it; otherwise, it is not.