If you'd like to hide the navigation links (including the search icon) in the Divi Theme's standard header, here are several options.

Hide the Divi Header Menu within Divi Settings

It's possible to remove the menu links and search icon within Divi itself. To do so:

1. Remove the search icon through the customizer, by disabling it at:

Divi > Theme Customizer > Header & Navigation > Header Elements > Show Search Icon

2. Remove all items from your menu, at:

WP Dashboard > Appearance > Menus

(Alternatively create a new blank menu and assign it to the "Primary Menu" display location.

Hide the Divi Header Menu Links using CSS

To hide the navigation links / search icon in the Divi header, you can add the following CSS:

Related Post: Adding CSS to the Divi Theme

Hide the Divi Header Menu Links using Divi Booster

Divi Booster contains an option to hide the Divi header menu links and search icon.

You'll find it in the Divi Booster settings at:

Divi > Divi Booster > 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, as described above. But the Divi Booster feature still works and you may wish to use it if you find it more convenient.