Will you call me to inform me that the courier is about to expire?

Generally (including telephone, SMS notification, etc. ), and the courier has the obligation to inform the recipient.

According to the Provisional Regulations on Express Delivery:

Twenty-fifth enterprises engaged in express delivery business shall deliver the express mail to the agreed address, the recipient or the agent designated by the recipient, and inform the recipient or agent to receive and send it in person. The consignee or agent has the right to face-to-face acceptance.