If you check the "Flip Mini-Map" box under more options, the chat box will stop displaying for the duration of that game. You can still press enter and type in it, and you teammates will still see what you type, and forfeiting still goes through... but you cannot see it displayed on your screen, for the chat box is completely gone. You cannot see your other teammates' messages, or other in-game messages, such as dragon kills or killstreak rewards.
If you start another game, it should work fine again, unless you check (or un-check) that box again. I wanted to return the minimap back to the other side of my screen after I died because I couldn't click on a portion of the in-game screen to run away from a gank (lol).
I think the chat box comes back if you flip it back to the original position where it was when the game itself started, and then it will go away again if you flip it back to the other side...
It'd be nice to not have to restart my client to get that change to go through successfully.
Attached is a dxdiag file, if that has anything to do with the problem.
I run the game on pretty much max settings with v-sync on.
I will follow this thread if more information about the issue is required. Thank you.
*EDIT* This bug was ALSO happening with the chat box BEFORE the Season 3 patch. Hope that helps!