I've solved my problem. It was caused by owner of the wp-content (chmod 755) dir and plugins dir. DBCR couldn't create cache dir in plugin dir (chmod 755) and couldn't copy files to the wp-content dir because on my web server PHP is running as "www-data" so there was dir owner mismath.
To solve this problem I've set temporarily chmod to wp-content to 777 and set chmod 777 plugins/db-cache-reloaded-fix next activated plugin and set back chmod 755 to wp-content and plugin dir and asked my admin to set the owner of the cache dir to "www-data" (the one which PHP runs as).
I think that taking ownership of wp-content dir by "www-data" user (for apache) would also do the trick but I've not tested it.
It workd for me. Hope it helps someone else.