field output dependend on mime type of another field
This can be done with a template preprocess added to the template.php file in your theme. You would need to test for the video mime type on the correct node and then use Drupal's hide function, to tell Drupal to skip the field on render.

    /**
     * Override or insert variables into the node templates.
     *
     * @param $variables
     *   An array of variables to pass to the theme template.
     * @param $hook
     *   The name of the template being rendered ("node" in this case.)
     */
    function THEMENAME_preprocess_node(&amp;$variables, $hook) {
      $type = $variables['type'];

      switch ($type) {

          case 'NODE_TYPE_MACHINE_NAME':

            if (isset($variables['FILE_FIELD_MACHINE_NAME'][0])) {
              $mime_type = $variables['FILE_FIELD_MACHINE_NAME'][0]['filemime'];
              $mime_regex = '/^video/';
              // We have to check the string for the presence of "video", because it could be any of 18 video mime types. 
              $is_video = preg_match($mime_regex, $mime_type);

              if ($is_video) {
                hide($variables['content']['IMAGE_FIELD_MACHINE_NAME']);
              }

            }
          break;
      }
    }