What is the ruling on an employee using a company car, which is fully insured and leased from a rental company, for personal use, provided that he bears the costs of fuel, oil, and washing, and without incurring any additional costs to the company? Is the permission of the responsible manager sufficient?

It is not permissible to use work tools for personal purposes except with the permission of the employer or whoever has the authority to grant permission. If your direct manager is authorized to dispose of the cars and permitted you to use them outside of work hours, provided you bear the fuel costs, then there is no harm in that if you work for a private company. However, in government institutions, it is not permissible to use their tools for personal purposes, even if the head permits it, because he does not possess that authority.