R. T. Mullins has a relatively recent article, Closeness with God: A Problem for Divine Impassibility, at the Journal of Analytic Theology, in which, as you might expect, he argues that closeness with God is a problem for divine impassibility. Nothing in the article is particularly new, or indeed all that interesting, but divine impassibility is in itself interesting, and it's worth pointing out the errors that Mullins makes in discussing the topic, in the hope that others avoid them.
We can begin to see the problems from Mullins's first characterization:
The majority of Christian theologians throughout history have said that God cannot be moved by creatures to feel anything. God does not literally have empathy, mercy, or compassion. Instead, God only feels pure undisturbed happiness. This view is called divine impassibility.
None of these are quite correct. For instance, most theologians who have accepted the doctrine of divine impassibility have held that "God has mercy", taken literally, is true. The obvious reason for it is that they would deny that mercy requires passibility. Empathy and compassion are somewhat more plausible, because the very names have elements (pathos, passio) that seem to imply passibility; however, on compassion, at least, the usual claim is that while compassion in us involves passibility, there is an activity involved in it that we can attribute to God, so "God has compassion" is also literally true, although God's compassion (perhaps unsurprisingly) is somewhat different from our own, because it lacks the particular aspect of passibility. It's a little odd to say that "God only feels pure undisturbed happiness" because happiness has historically not been seen as a feeling at all -- the view that happiness is a feeling only became dominant in the past two hundred years or so. The classical view is that we don't know what God's 'inner emotional life' is like, because we are not God; but in any case, if we are going to call how God acts toward us 'happiness', there are lots of other things we can legitimately call it, as well -- including 'mercy', 'love', and so forth.
The first sentence admits of a correct interpretation, but the natural reading of it leaves the impression that the doctrine of impassibility is about feeling. This is not really the focus of the doctrine. Passibility is not primarily about feelings, although our feelings are tied to our passibility, which is why some of them are called passions. Passibility is about being such that you must undergo things. God, however, is purely active and has no potentiality; He can't undergo anything. As I said, in our own case, some of the ways we are passible are called passions, for precisely the fact that they are things we are forced to undergo by the world around us. If I come up to you and slap you in the face, you will find certain effects forced on you, over which you will have only limited control: surprise, pain, anger, bewilderment, and so forth. But it need not be feelings. If I shine a light directly into your eyes, for instance, you will be forced to undergo it and to react it. The doctrine of divine impassibility says that nothing can be forced onto God like this. One reason why it has been so common to think of God as impassible is that there is very good reason to think that passibility is a sign of destructibility. The passibility of your visual system means I can blind you by overpowering it; the passibility involved with pain and passions is what makes torturing people and breaking them possible. The physical passibility of a body means it can be torn apart. Everywhere we look, things are destructible in the way they are passible. But God obviously is not destructible; God cannot be overpowered; God cannot be forced in any way, and therefore cannot be forced to undergo anything; in fact, God doesn't undergo things, because everything other than God presupposes His action, not vice versa.
You'll notice that I keep saying 'passions', not 'emotions'. These terms are often thrown together today, but historically they would not have been, for the obvious fact that etymologically they are opposites. 'Passion' literally means you are moved, stirred up; emotion means you move something, stir it up. 'Passion' suggests a kind of passivity; 'emotion' suggests a kind of activity. In contemporary English they have become muddled together, but it's obviously going to be relevant here, since while the doctrine of divine impassibility rules out God having 'passions' in the original sense, it doesn't rule out God having 'emotions' in the original sense. Mullins correctly recognizes the latter, but fails to consider that this could indicate a problem that the field of things being talked about is muddled and not entirely coherent, due to the historical mingling of two very different ways of talking about human interaction with the world.
When Mullins tries to characterize empathy, he does so as follows:
EMPATHY: Sally empathizes with Ben if and only if (i) Sally is consciously aware that Ben is having an emotion E, (ii) Sally is consciously aware of what it feels like to have E, and (iii) on the right basis Sally is consciously aware of what it is like for Ben to have E.
On the basis of this, Mullins argues that the doctrine of divine impassibility implies that only (i) could be true of God. (ii) can't apply, because God doesn't know "what it feels like" to experience misery because He only feels happiness and because God cannot know "what it is like" to experience emotions in a way that depends on external things. (iii) can't apply, because God can't be influenced by things other than Himself and therefore can't experience others.
Whether or not Mullins's characterization of empathy is a good one, Mullins is incorrect; (ii) and (iii) of his characterization can apply to God. God knows what it feels like to experience misery, and He knows what it is like to experience emotions forced on one by external things, not because He has experienced these things but because He invented the experiences. They only exist because He conceived of them and created beings who could have them. What is more, our knowledge of these things is vague and dim and sensory-based; God's knowledge has none of these limitations, so He knows these things better than we do. The mistake made here (common among passibilists) is assuming that you can only know feelings and emotions by experiencing them; that is at least more or less true of us (although it's questionable whether it's always true even of us), but there is no reason to think it is true of the omniscient Creator who made us, and without whose fully knowledgeable creation none of those feelings would exist at all.
The reason that impassibilists have tended to hold that 'empathy' can apply only metaphorically to God is that, unlike compassion, which involves a much more complicated range of behavior and is used in a much wider range of contexts, 'empathy' seems in its normal usage specifically to highlight undergoing the same thing as someone else. If one interprets Mullins's account of empathy as requiring this, then 'empathy' in Mullins's sense could only apply to God metaphorically. But the characterization doesn't actually say this. (ii) is put entirely in terms of 'conscious awareness', which in his arguments he treats as a kind of knowledge; but God has knowledge. He has the most and best knowledge of everything. What's more, as classically understood, God's knowing something is a precondition for anyone else knowing it; it's just that under impassibility He knows things without having to undergo them, because His knowledge is a precondition for their existence. So nothing in (ii) requires that 'empathy' in Mullins's sense be applied to God only metaphorically. The 'on the right basis' in (iii) might be taken as requiring some sort of undergoing; except that Mullins also characterizes it in terms of knowledge, which God has in superabundance, by nature and perhaps by will, not by being forced to endure things the way we sometimes are. "God does not need to learn anything from the school of hard knocks" is not the same as saying "God does not know what we only know from the school of hard knocks."
Mullins of course does the usual bait-and-switches that passibilists have historically loved ("Ooh, God sounds like a psychopath!"), which can be dismissed as childish nonsense, particularly since he's failed to formulate an account of empathy that doesn't apply literally to God, and more than that, because He never bothers to consider the obvious next question even if it didn't, which would be whether 'empathy' can be applied metaphorically to God in a way that is relevant to this question. Like a great many people, Mullins seems not to grasp that when we speak metaphorically, we nonetheless say meaningful things, and therefore if a term applies not literally but metaphorically, that doesn't dismiss the term but just changes the way we approach investigating it. He also says that there is "zero evidence" of impassibility in the Bible, but this is just because he has a bad habit of assuming that if there's a thesis about God that he doesn't like that it must have just been made up randomly by some people for no reason; he never bothers to ask what people were reflecting on that led to their acceptance of it in the first place. In the case of impassibility, it's what it means for God to be Creator, what it means for God to have foreknowledge, what it means for God to be everlasting, what it means for God to be sovereign over all, all of which can be supported by Scriptural evidence. Mullins no doubt thinks that the evidence has been misinterpreted, but to claim that it isn't there is merely a disingenuous attempt to make his argument easier.
After all, passibility is not directly stated in Scripture, either; it requires interpreting feeling-terms applied to God as both (a) applied in a sense that requires passibility and (b) applied literally. That neither of these can just be assumed can be seen from the many physical expressions and the many obvious metaphors that Scripture uses to talk about God, and from the fact that many of these usages are clearly emphasizing not divine passiveness but divine activity. Mullins wants to argue that there is a sharp and obvious distinction between physical and 'emotional' cases, but this is simply false; a very large portion of the latter occur in contexts where the former is occurring, also. The emotional expressions are very often associated with the divine face, the divine arms, the fire of God's breath, and so forth. The impassibilists can accept all these passages; they just hold that in each case the terms are either used in ways that don't imply passibility or that they are used as metaphors to describe divine actions. Describing something by a metaphor doesn't imply that you don't think it exists; if wrath, compassion, patience, etc., are applied to God metaphorically, they are describing something, and what is more, one could perfectly well say that the metaphors are the best ways we have in our language to describe them. We just can't assume that they have all the baggage they would bring if they were used literally. This is nothing particularly new; we often have to do this even with human beings, which is why so much of our language for describing people is metaphorical. And becomes more and more so when we talk about saints, and heroes, and geniuses, who seem sometimes to be a little less passible in their knowledge, or will, or character, than the rest of us. How much more would this have to be the case for God?
And likewise there can be no problem with 'closeness to God'. God knows our feelings better than we know them ourselves, because He knows us better than we know ourselves. All our knowledge of any person in our lives is but a dim and wavering shadow compared to what God knows of us. His love is impassible and can never be overpowered, never turned aside. The difficulty in the relationship is all on our side. And why? Because our love is passible, our minds are passible, our bodies are passible, and thus we can waver, get knocked off course, get overpowered, break down, when it comes to matters of knowledge and love. And how is this problem dealt with? By becoming less passible. By virtues, which steady us and strengthen us, and by relying on God's greater (because wholly impassible) reliability.