Add Div to Comment Form

If you want to add some custom HTML between the </h3> and the <form> tags, you can try the following:

/**
 * Add custom HTML between the `</h3>` and the `<form>` tags in the comment_form() output.
 */
add_action( 'comment_form_before', function(){
    add_filter( 'pre_option_comment_registration', 'wpse_156112' );
});

function wpse_156112( $comment_registration )
{
     // Adjust this to your needs:
     echo '<div>Custom HTML between the closing H3 and opening FORM tags.</div>'; 

    remove_filter( current_filter(), __FUNCTION__ );
    return $comment_registration;
}

where the output is:

...</h3>
<div>Custom HTML between the closing H3 and opening FORM tags.</div>
<form>...

Otherwise you can use the comment_form_top action to add custom HTML right after the opening <form> tag:

/**
 * Add custom HTML just after the opening `<form>` tag in the comment_form() output.
 */
add_action( 'comment_form_top', function(){
     // Adjust this to your needs:
     echo '<div>Custom HTML just after the opening FORM tag.</div>'; 
});

with the following output:

<form>
    <div>Custom HTML just after the opening FORM tag.</div>
    ...

Here you can see the comment form of the TwentyTwelve theme with both methods activated: