Shortcode called twice

Shortcodes should never echo; they should always return the text to be displayed. See the User Contributed Notes in the add_shortcode() docs.

Your code should read more like this:

function displayTable() {
    $string = '';
    $string .=  '<table>';
    $string .=  '<tbody>';
    $fields = get_field_objects();
    foreach($fields as $field)
    {
      $string .=  '<td>';
      $string .=  '<td>'. $field['label'] .'</td>';
      $string .=  '<td>'. $field['value'] .'</td>';
      $string .=  '</tr>';
    }
    $string .=  '</tbody>';
    $string .=  '</table>';
    return $string;
}
add_shortcode('popnagel-tabel', 'displayTable');