Is there a keyboard command for “Switch to next open document”? (Ctrl+F6 in MS Word, Shift+F3 in WordPerfect.)
It depends on your operating system. Indeed, there is no built-in way to do this with keyboard in LO.
In LO, you can do this by clicking the Window menu and then choosing your opened document. Every LO window is in this list, no matter this is a Writer, Calc, Impress, etc.
With the keyboard, you should use the built-in tools of your operating system. Alt+Tab is very generic, the only problem is that it switches between all opened windows (even those which are not LO windows).
In Unity (and apparently Gnome Shell), you can use Alt+² (the key above Tab).
On Windows 7, it seems there is a way too.
An inclusive list of Windows short-cuts can be found on the company documentation. You can also install a dedicated program to do like Alt+² in Unity.
Anyway, this is something that should be implemented at the OS or graphical environment level, to ensure constant behaviour through various programs.
It’s nice to be able to toggle between open documents with gnome3 using Alt+² (the key above Tab).
Thanks for the tip.
Awesome. How in the world did you find that!! 
It works under KDE as well.
THANK YOU.
Alt+~
Also works in Ubuntu 18.04
Thank you. This is what I suspected. Using only keyboard, it looks like the closest is Alt+W then manually go down the list to the desired document.
Yes! At least in Linux there is … option - ` (the key above the TAB key) then use arrow key to selection the document to switch to. (I am using a mac keyboard so things may be confusing … if option key doesn’t work try the alt ` key instead …)