To hide the navigation links (including the search icon) in the Divi Theme's header, you can add the following CSS:
Related Post: Adding CSS to the Divi Theme
This is available in Divi Booster under “Deprecated > Divi 2.3 > Hide header links and search”. Note that it’s in the “Deprecated” section as it’s now possible to achieve the same effect within Divi itself. To do so, you can hide the search icon through the customizer and remove all items from the menu. But the Divi Booster feature still works and you may wish to use it if you find it more convenient.