What is the ruling of Islamic law regarding a department head’s refusal to approve paid leave requests for working from home for employees who can only perform their tasks from the workplace, and would he be committing a sin if he were to approve such requests at the behest of his manager?

If the situation is as you described, then you are not permitted to authorize workers to stay in their homes, following the example of other departments. And if your manager orders you to allow them to stay, you are not permitted to obey him unless he is authorized by the company owners to grant them permission.