Can I run tasks when orders are printed in Order Printer?
As of this writing, Shopify's Order Printer app doesn't support tagging, or webhooks, or any other method of recording when an order has been printed. This means that apps like Mechanic have no way of responding when an order is printed using that app.
For an alternative, consider OrderlyPrint. This app does support tagging orders, which means that you can run Mechanic tasks in response to orders being printed.