make entire tooltip clickable to event's permalink - place file at [your-theme]/tribe-events/month/tooltip.php

tooltip.php

<?php
/**
 * Please see single-event.php in this directory for detailed instructions on how to use and modify these templates.
 *
 * Override this template in your own theme by creating a file at:
 *
 *     [your-theme]/tribe-events/month/tooltip.php
 *
 * 2017-03-14 for https://theeventscalendar.com/support/forums/topic/how-to-make-featured-image-in-tooltip-clickable/
 * from https://gist.github.com/cliffordp/cbce6b3b8eae3dba89faf4d83f596582
 * Note: valid HTML 5 but not HTML 4. See http://stackoverflow.com/a/1828032/893907 for details.
 *
 * @version 4.4
 */
?>

<script type="text/html" id="tribe_tmpl_tooltip">
    <a class="tribe_tmpl_tooltip-custom-link" href="[[=permalink]]">
	<div id="tribe-events-tooltip-[[=eventId]]" class="tribe-events-tooltip">
		<h4 class="entry-title summary">[[=raw title]]</h4>

		<div class="tribe-events-event-body">
			<div class="tribe-event-duration">
				<abbr class="tribe-events-abbr tribe-event-date-start">[[=dateDisplay]] </abbr>
			</div>
			[[ if(imageTooltipSrc.length) { ]]
			<div class="tribe-events-event-thumb">
				<img src="[[=imageTooltipSrc]]" alt="[[=title]]" />
			</div>
			[[ } ]]
			[[ if(excerpt.length) { ]]
			<div class="tribe-event-description">[[=raw excerpt]]</div>
			[[ } ]]
			<span class="tribe-events-arrow"></span>
		</div>
	</div>
    </a>
</script>

<script type="text/html" id="tribe_tmpl_tooltip_featured">
    <a class="tribe_tmpl_tooltip_featured-custom-link" href="[[=permalink]]">
	<div id="tribe-events-tooltip-[[=eventId]]" class="tribe-events-tooltip tribe-event-featured">
		[[ if(imageTooltipSrc.length) { ]]
			<div class="tribe-events-event-thumb">
				<img src="[[=imageTooltipSrc]]" alt="[[=title]]" />
			</div>
		[[ } ]]

		<h4 class="entry-title summary">[[=raw title]]</h4>

		<div class="tribe-events-event-body">
			<div class="tribe-event-duration">
				<abbr class="tribe-events-abbr tribe-event-date-start">[[=dateDisplay]] </abbr>
			</div>

			[[ if(excerpt.length) { ]]
			<div class="tribe-event-description">[[=raw excerpt]]</div>
			[[ } ]]
			<span class="tribe-events-arrow"></span>
		</div>
	</div>
    </a>
</script>