I know this thread is 2 years old but it didn’t have the solution I was looking for, so I did some digging and figured out how to remove the Search button entirely. (So I’m putting it here in case anyone else looks for the same thing.)
CSS will work to hide the Search button, but you can also remove it completely by modifying the bbPress template files.
It took me a long time to find the right one, but what you wanna do is go to your plugins folder, and find the bbpress folder. Then look inside bbpress > templates > default > bbpress, and find the file named ‘content-archive-forum.php’
This file outputs the <div id="bbpress-forums"> code at the top of your index page, then immediately after that it outputs the search field and the breadcrumbs.
To remove the search box ONLY from your Forum Index page, just comment out the following lines of code:
/* Comment these lines out to remove search from your Forum Index page.
if ( bbp_allow_search() ) : ?>
<?php bbp_get_template_part( 'form', 'search' ); ?>
<?php endif; */ ?>
<?php bbp_breadcrumb(); ?>
This will ONLY disable the search box; if you want to remove the breadcrumbs as well, then you need to comment out (or delete) the bbp_breadcrum(); line as well.
NOTE: The safest way to do this is to create a bbpress folder inside your theme, and save the modified content-archive-forum.php there (otherwise it’ll be overwritten the next time bbPress updates.)