How to remove specific categories from tag archives？

One way to do this would be to create a custom widget which displays categories using wp_list_categories(). Then exclude any categories you don’t want displayed using the exclude option of wp_list_categories(): http://codex.wordpress.org/Template_Tags/wp_list_categories#Include_or_Exclude_Categories.

This example below creates a custom categories widget in Appearance / Widgets. This code can be added to functions.php or used in a plugin.

/*-----------------------------------------------------------------------------------*/
/* Create category list widget */
/*-----------------------------------------------------------------------------------*/
class ABCD_Category_Widget extends WP_Widget
{
  function ABCD_Category_Widget()
  {
    $options = array( 'classname' => 'ABCD_Category_Widget', 'description' => 'Displays a list of categories' );
    $this->WP_Widget( 'ABCD_Category_Widget', 'ABCD Category List', $options );
  }

  function form( $instance )
  {
    $instance = wp_parse_args( ( array ) $instance, array( 'title' => '' ) );
    $title = $instance['title'];
  ?>
  <p><label for="<?php echo $this->get_field_id( 'title' ); ?>">Title: <input class="widefat" id="<?php echo $this->get_field_id( 'title' ); ?>" name="<?php echo $this->get_field_name( 'title' ); ?>" type="text" value="<?php echo attribute_escape( $title ); ?>" /></label></p>
  <?php
  }

  function update( $new_instance, $old_instance )
  {
    $instance = $old_instance;
    $instance['title'] = $new_instance['title'];
    return $instance;
  }

  function widget( $args, $instance )
  {
    extract( $args, EXTR_SKIP );

    echo $before_widget;
    $title = empty( $instance['title'] ) ? ' ' : apply_filters( 'widget_title', $instance['title'] );

    if ( !empty( $title ) )
      echo $before_title . $title . $after_title;

    $args = array(
    'orderby'       => 'name', 
    'order'         => 'ASC', 
    'number'        => null,
    'optioncount'   => true, 
    'exclude_admin' => true, 
    'show_fullname' => false,
    'hide_empty'    => true,
    'echo'          => true,
    'style'         => 'list',
    'exclude'       => array( 20, 26 ), // IDs of categories you want to exclude
    'title_li'      => __( '' ),
    'html'          => true
    );


    echo '<ul>', wp_list_categories( $args ), '</ul>';

    echo $after_widget;
  }

}
add_action( 'widgets_init', create_function( '', 'return register_widget("ABCD_Category_Widget");' ) );