I don't know about now since I don't visit 4chan anymore, but the imageboards very clearly used a DB - during periods of high load one of the most common error messages was "MySQL connection error" which became a sort of minor meme for a while (this was several years ago.)

On the other hand the textboards do appear to not use a traditional RDBMS - they were running a modified version of Shiichan (http://wakaba.c3.cx/shii/shiichan ); Kareha (http://wakaba.c3.cx/s/web/wakaba_kareha ) is another popular textboard script that doesn't need a database.

I remember reading that 4chan saved each page as just that, static pages. The server saves to each page when a new comment is posted to the relevant page and serves up pages based on last modified times.

AFAIK sites like 4chan store post data in a database then use that to rebuild the static html files whenever someone posts or deletes.