Does it mean there's a newspaper in that case? Should they go away after I've found them?

No, it means you found a newspaper in that case already. It only shows ones that you already found.

I think it shows ones you dont have yet either.  I beat the game and missed 1 newspaper and on my case files it has a darkened out newspaper.  I assume that is the one I missed.  The ones you have found are not darkened out.
 
Perhaps they only show up once you have beaten the game tho...I dunno.

The game tells you what cases have newspapers in them, regardless if you have them or not. If you completed the case and the newspaper is grayed out, that means you didn't find it.