Hide custom field if empty

You’re not running check on whether the artist’s instagram field is empty, you’re running a check on whether the ‘artists’ term is empty, which it isn’t.

<?php
    $artistigs = get_the_terms( get_the_ID(), 'artists' );
    if( ! empty( $artistigs ) ) : ?>
         //This will execute if the artist term is not empty
    endif;
?>

I don’t use ACF myself so not 100% sure how stuff like the_field() works but try this:

<?php
    $artists = get_the_terms( get_the_ID(), 'artists' );
    if( !empty( $artists ) ) : ?>    
    <?php foreach( $artists as $artist ) :
        $artistig = get_field( 'artist-instagram', $artist );
        if( !empty( $artistig ) ) :
            echo '<div>';
                echo '<i class="fab fa-instagram"></i> <strong><a href="https://instagram.com/' . $artistig . '" target="_blank">@' . $artistig . '</a></strong>';
            echo '</div>';
        endif;
   endforeach; 
  endif;
?>

================

UPDATE

To address your update, the var_dump() is really going to mess with formatting, so lets just remove it.

<?php 
    $artists = get_the_terms( get_the_ID(), 'artists' );
    if( !empty( $artists ) ) :  
        foreach( $artists as $artist ) :        
            $artistig = get_field( 'artist-instagram', $artist );
            if( !empty( $artistig ) ) : 
                echo 'IG User TEST: '. $artistig .' END TEST';
            endif;      
        endforeach;
    endif; 
?>

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