Include HTML (Bootstrap Modal Box) with a plugin into my header

Question

You only need the_content filter to add the modal, but bootstrap.js is needed to make it work.

add_action('wp_enqueue_scripts', array($this, 'enqueue_bootstrap');

public function enqueue_bootstrap(){
    wp_register_script( 'bootstrap', plugins_url( 'your-plugin/assets/js/bootstrap.min.js' ) );
    wp_enqueue_script( 'bootstrap' );
}

EDIT:

You need to add $content argument to your function.
As the codex says

Note that the filter function must return the content after it is finished processing, or site visitors will see a blank page and other plugins also filtering the content may generate errors.

in your case:

public function dmd_fav_modal_box($content) {

    $modal="<div class="modal fade" id="dmd_favorite_modal" tabindex="-1" role="dialog" aria-labelledby="dmd_favorite_modalLabel">
      <div class="modal-dialog" role="document">
          <div class="modal-content">
              <div class="modal-header">
                  <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">&times;</span></button>
            </div>
            <div class="modal-body">
                <p>Test</p>
            </div>
            <!--div class="modal-footer">
                <button type="button" class="btn btn-default" data-dismiss="modal">Close</button>
                <button type="button" class="btn btn-primary">Save changes</button>
            </div-->
        </div>
    </div>
</div>";
return $modal.$content;