As panjala Srinath wrote, the later versions of Tableau can use JSON files as data sources. That's a big improvement. However, that seems to be the limit of its JSON support.
I use Tableau to connect to databases that contain JSON strings in some columns. Tableau doesn't seem to have any functions for parsing those strings. I've seen some people on this forum request a parsing feature, but there doesn't seem to be any activity. Other people have tried using regular expressions to get data out of JSON, but that's not always effective.
I'd like to see Tableau, the company, be serious about this and add good JSON support to the application.
I have been struggling with nested (having arrays) JSON files which aren't readily supported by Tableau till date. Looking forward to witnessing this becoming a reality however.