Hey there, greenplants.
Thanks for bringing your question forward here in the Community.
While there are many options concerning invoices in QuickBooks Online, there isn't currently a way to mark them as printed. I can see how having this ability would be useful to you and other QBO users. I recommend submitting feedback to the Product Development Team for review. This can be done directly through your account by clicking the Gear icon in the top right corner, then select Feedback. This is the best way to make your requests known as many product updates and feature releases come directly from great user suggestions like yours.
Once you've entered your feedback keep an eye out on the Firm of the Future site for info about upcoming changes to the software.
If you have any other questions or concerns in the meantime please don't hesitate to reach back out in the comments below.