As a business, I offer an express shipping option for $10 along with free standard shipping.
Do I have to refund express shipping cost which customer paid in case of a warranty claim?
An example would be customer claims warranty after 5 months and I can't fix the fault. Does this mean I have to refund $10 shipping cost as well?
Express shipping is a service separate to the product. Your client used that service.
However, for good will I would refund that $10 too. If you don't, you risk your client telling everybody they know that you are dodgy.