BuddyPress.org

As @rogercoathup suggested, you should open up header.php in your child theme, find the code that creates the Activity tab, and delete it.

If you don’t yet have header.php in your child theme, you’ll have to copy it here. More on BP child themes: https://codex.buddypress.org/extending-buddypress/building-a-buddypress-child-theme/