If a customer pays for the same invoice twice, should the customer be informed?

I say yes. If you become aware of the double payment when posting the customer's second remittance, I would double check your records to be certain you are not owed money from the customer and would then inform the customer.  If the check was sent in error, I would photocopy the check, document on the photocopy what had occurred, and then return it to the customer.

If you did not notice the double payment when processing the customer's remittance, the customer's accounts receivable record will show a negative amount due for the sales invoice and might even show a negative amount due from the customer. If your company mails statements to its customers, the customer should be able to see its double payment when reviewing the statement.