You can do it pretty easily with jQuery. You'll want to add a class to your tag so you can select it:
<a class="toggle-link" data-toggle="collapse" data-parent=".event-list-links" href="#collapse-<?php the_ID(); ?>">
<i class="icon-chevron-sign-down"></i>
</a>
Then just add a click handler that removes the icon-chevron-sign-down class and adds a icon-chevron-sign-up class.
$'.toggle-link').click(function() {
$(this).find('.icon-chevron-sign-down').
removeClass('icon-chevron-sign-down').
addClass('icon-chevron-sign-up');
return false;
});
I'd note that this won't switch back to icon-chevron-sign-down if you click again. That code is a little more complicated. Let me know if that's what you're looking for and I'll update this answer.