LOL nice question! (It has me imagining putting a heavy old CRT monitor face-down on a copier, and having it break through the glass top of the copier.)
I’d be interested to see the results. I think it would sort of work but might look weird. Usually copiers and scanners that I’ve known use their own light and focus on the surface, expecting something like a piece of paper, so depending on the copier, I think it would either come out about as you might hope but maybe a bit blurry, or possibly more interesting than that, or possibly not work well at all.
As long as your monitor is light enough and it’s easy enough to do, I’d try it out and see, and hopefully scan an example and share so we can see.
Of course, if you want a perhaps-more-practical (or at least conventional) solution, you might try either taking your laptop (or the content on a USB drive) to a library or copy center.