How to delete an old Apple Account from my iPhone?

It depends on which screen you are seeing and what you are doing where you see that.

• If an app is trying to update that was downloaded with the previous Apple ID, then all you need to do is delete the app.
• If the device is currently signed into the previous Apple ID and you need the password to sign out, then if you cannot gain access to that account, you will need to provide the original purchase receipt to remove the Activation Lock.How to remove Activation Lock - Apple Support
• Since you are being asked for security questions, it would indicate that you do not have Two Factor Authentication turned on, which would make resetting the password easer. With Two Factor Authentication, Account Recovery is available here: How to use account recovery when you can’t reset your Apple Account password - Apple Support