I found this to be some fantastic background history on the ideas of impassibilty/non-impassibility by Alister McGrath:
The Q/A on God’s suffering is question 4, about halfway down.
‘Underscoring this question is another question before moving to answer the question of patripassianism or theopaschitism. This is the priority of Hellenistic philosophy (metaphysics) over the biblical understanding of God. The issue of whether God can suffer opens this question to scrutiny. Adolf von Harnack’s Dogmengeschichte theorized that many central Christian doctrines were influenced by non-biblical worldviews and that one could uncover the ‘fossilized’ doctrines through a method of redaction. One such issue is the imposition of the dualistic Greek notion of an ‘impassible’ understanding of deity which is removed from human passions (equated with materialism). Thus, cuing more from Plato’s perfect and unchanging (as passions were changeable) ‘deity’ of the Forms, Christianity came to understand God as unchangeable and thereby perfect. God, in short, could not change nor be moved from perfection lest in doing so God becomes less perfect and thereby, by definition, no longer divine. To suffer would be to ‘feel’ change, and immutability of substance or will becomes confused with immutability of experience. In Philo through to Aquinas, the thought that God could suffer would mean nothing less than the fact that God could be altered, by compassion or love, by the experience. The prime mover and un-moveable is moved or changed by human predicament or experience of that predicament. God’s compassion, a biblical motif, is naught but figurative and not an attribute or predication of deity itself. Harnack felt that this type of dogma was due to the imposition of alien metaphysics rather than the exposition of the Hebrew and Christian bible. Many believe, at some level, him to be correct.’