Doing action based on input from options menu

You should post to the same URL and catch the submit ex:

function option_page_html(){
    if (isset($_POST['plugin_action']) && $_POST['plugin_action'] == 'post_url'){
        include_once(plugin_dir_path(__FILE__).'post-video.php');
    }
    ?>
    <div>
        <form action="" method="post">
            <p><h3>Paste video url</h3></p>
            <input type="text" size="100" name="url" /><br/>
            <input type="hidden" name="plugin_action" value="post_url">
            <input type="submit" value="Post"/>
        </form>
    </div>
    <?php
}

note : this is an example and you should make sure you sanitize and validate the value and probably use nonce for better security