I used to change X11 desktops in linux and then move back to whichever one was running Warcraft and tab back to it. This worked fine for quite awhile but stopped working a few updates back as the game would crash when I attempted to return to it.
I've found a workaround if you want to enable similar functionality. Within Warcraft you have to go to System --> Graphics --> Display Mode --> set to Fullscreen (Windowed)
This changes the behavior somewhat from simply setting it to Fullscreen without the windowed element. With windowed Warcraft will continue to render the whole time it's running even if it's on another desktop. Without windowed Warcraft will go to the background while it's not in focus and cease rendering. I'm running on Intel graphics. While I wouldn't mind hearing if anyone else is having or not having this issue in the first place this post is mainly to share the info above in case it might help someone else.