The conditional logic is difficult to understand.

You want to display an icon (fontAwesome based) for comments – only availabe for logged in users who meet the following requirements:

1. is not the 'designer' of this post.
2. has the role 'customer'

Not sure, your conditional logic is set up properly as it seems to output the comment markup for any user who is not the 'designer' ... pardon me – but where's the difference between author_id and designer_id?

A slightly more DRY implementation might be this code:

<?php
$current_user_id = get_current_user_id();
$designer_id = get_the_author_meta("ID");
$author_id = '???';
$show_comment_icon = false;

if (is_user_logged_in() && $current_user_id != $designer_id ) {
    // if user has role "customer" 
    if (in_array ('customer', (array)$user->roles)) {
        // if currently logged in user is the author of the current post?
        if ( $current_user_id == $author_id) { 
            $show_comment_icon = true;
        }
    } else { 
        // if user is not a "customer" 
        $show_comment_icon = true;
    }
 
    // comment icon html template
    $comment_icon = 
    '<i 
        class="fa fa-comments send_designer_msg" 
        data-designer-id="'.$designer_id.'" 
        data-logo-number="'.$entry_no.'" 
        aria-hidden="true">
    </i>';  
    
    // output your html
    if($show_comment_icon){
        echo $comment_icon; 
    }       
}
?>

Even though this revised code snippet certainly flaws: Some recommendations to avoid duplicate code and improve readability in php:

• choose an appropriate enclosing concept (when switching beween php and html)*
• seperate 'data compilation' and logics from output
• insert comments in your code generously
• save recurringly queried values (e.g returned by functions/classes) in variables
• make the effort to revise your code in the end (probably no one is rewarding you for this effort – besides yourself, to be able to understand your code even after a few years)

enclosing/switching beween php and html code Your code is valid. However, legibility suffers. A feasable rule of thumb to decide whether to use rather enclosed php-in-html or or html-in-php is the proportion.

If there's more php processing involved (conditions, calculations, filtering etc.) better use a variable to define your html output.

If the html template part is predominant you could use something like this (similar to the default wordpress loop proposal):

<?php if($condition_fullfilled) :?>
    <p>Condition is fullfilled – there is a lot more html to come!</p>
    .... (200 lines of html markup)
<?php else: ?>
    <p>Nothing found, matching your conditions</p>
<?php endif; ?>