If you'd like to turn off the Automatic Payment Notifications, you can...
Head into your account settings by clicking your name in the top right, then clicking Account.
Next click 'Settings', then 'Notifications'
From here you can disable Payment Due Reminders.
You are able to enable/disable these reminders project by project by clicking the 'Gear' symbol on your projects.
If you've disabled Payment Reminders on the account level, it will disable for all existing projects.