That can be a bit difficult to track. How are you printing invoices? How do you track whether your customer print an invoice you send attached in an email?
Let say you have a way to track this, I would create a custom order attribute to flag orders and show it in a custom column in grid, look at this for example: Mage_Sales_Controller_Abstract::printAction()
to a place where set the attribute to true.
You can have your custom controller to handle your attribute flag and then redirect to Mage_Sales_Controller_Abstract::printAction()
For more examples on printActions look in admin controllers at:
app/code/core/Mage/Adminhtml/controllers/Sales/Order/
If you are using print functionality from backend then you can add the flag attribute and set the attribute by listening one of the events controller post/pre dispatch. Then show attribute in grid.
This is a good resource on how to add custom attributes to orders