The fourth parameter to wp_set_object_terms()
is an “append” argument.
$append
(bool) (required) If true, tags will be appended to the object. If false, tags will replace existing tagsDefault: False
http://codex.wordpress.org/Function_Reference/wp_set_object_terms
Your code should work if you pass true
as that fourth argument.
wp_set_object_terms(
$post->ID,
explode( ',', $_POST['postTags'] ),
'product_tag',
true
);