Try this:
<marquee onmouseover="this.setAttribute('scrollamount', 0, 0);" onmouseout="this.setAttribute('scrollamount', 2, 0);" behavior="scroll" width="100%" loop="infinite" direction="up" scrollamount="2" scrolldelay=".0001" height="300">
<?php
$the_query = new WP_Query( 'showposts=10' );
$now = current_time('timestamp');
$still_new = 60*60*24*10; // 10 day window
while ($the_query -> have_posts()) : $the_query -> the_post();
if ( $now - get_the_time('U') > $still_new ) { ?>
<li class="noticboard"><h3><?php the_time('d/m/Y'); ?> </h3> <p> <a href="https://wordpress.stackexchange.com/questions/204942/<?php the_permalink() ?>"><?php the_title(); ?></a></p> </li>
<?php } else { ?>
<li class="noticboard"><h3><?php the_time('d/m/Y'); ?> <img src="wp-content/plugins/scrolling-notice-board/icons/new.gif" /></h3> <p> <a href="https://wordpress.stackexchange.com/questions/204942/<?php the_permalink() ?>"><?php the_title(); ?></a></p></li>
<?php } ?>
<!--<li><?php echo substr(strip_tags($post->post_content), 0, 100);?></li>-->
<?php endwhile;?>
</marquee>
It uses the difference in the unix timestamp between now and the date of the post to calculate whether the post is 10 days old or not.
$still_new = 60*60*24*10; // 10 day window just allows you to set a window (10 days here) during which the ‘new’ image will be shown.