What controls the message displayed at the bottom of invoices?

"This invoice is due on MM/DD/YYYY".

This message reflects the due date of an invoice, which in turn is controlled by the payment terms set for the project. The terms can be set from Settings > Billing & Invoices > Accounts Receivable, or at the client and project level from the respective screens (Details > Billing Options tab). However, the setting at the project level overrides the one at the client and global level.