Code to allow to create and update a user in WordPress without an e-mail address. It simply removes the error message. This code was written to be used in a plugin written as a class

gistfile1.php

public function __construct () {
    // other code...
    
    add_action( 'user_profile_update_errors', array( $this, 'remove_empty_email_error' ));
}

function remove_empty_email_error( $arg ) {
    if ( !empty( $arg->errors['empty_email'] ) ) unset( $arg->errors['empty_email'] );
}