You can set contract notifications by going to Setup -> Settings -> Cron Job -> Contracts.
Only the contract creator or admin will receive an expiration notification for a specific contract.
If a contract is expired, the list table row will be filled with a red background. You have the option to receive contract expiry notifications in addition to email. Check the email template from Setup -> Email Templates.
If you don't want to receive emails for contract expiration notifications, you only need to disable this email template.