The "toggle visibility" button in the top right corner of shoutbox (a minus sign) should do the job.
The shoutbox is not completely removed, but collapsed to just the heading. (the button will become a plus sign to make it visible again).
I presume cookies must be enabled for this...
...you have to?
For me the setting is kept, so it stays "collapsed" (invisible).
The cookie responsible for that is "xf_siropuShoutboxCollapsed".
Mine has the value set to "true", so xf_siropuShoutboxCollapsed:true.