Can I disable the Steam Community notification in Big Picture Mode with PS4 controller playing Diablo 2
You have two options.
First involves turning steam community off (ref: https://gaming.stackexchange.com/a/24676/1770):
Either notifications need to be turned off globally (View -> Settings -> Friends -> Uncheck all the "Display a Notification" check boxes), or the steam community needs to be disabled in-game in order to prevent me from receiving notifications in-game.
The second option is to install DS4Windows (http://ds4windows.com/) and play with button mappings to see if you can map "Shift + Tab" to the controller