I would use the woocommerce_after_single_product hook. Then use get_the_tags().
Something like this within your functions.php of your theme:
add_action('woocommerce_after_single_product', 'add_tags_to_product');
function add_tags_to_product() {
$posttags = get_the_tags();
if ($posttags) {
foreach($posttags as $tag) {
echo $tag->name . ' ';
}
}
}