Does the Bible indicate that God is subject to sudden, involuntary, non-intellectual upheavals in his usually calm state of mind? Well, hardly. The Westminster Confession, the best summary of the contents of the Bible, says that God is without parts or passions. Parts refers to bodily organs. Bodies have parts, minds do not. But God is also without passions. The word passion, in more modern terminology affection, is wider than the term emotion but includes the latter. A passion or affection is the result of being affected by some external force. A dog is affected by a whipping; a student is affected, sometimes, by the possibility of a good grade. There are modern psychology books written about “the affective consciousness.” But God is not affected by anything. Of, in another translation of the Greek term, God does not “suffer” anything.
On the contrary, not only the Westminster Confession, but all or nearly all the historic creeds says that God is immutable. He does not change. Emotion, however, is a sudden, involuntary change. To have emotions would be inconsistent with God’s eternal state of blessedness.
Now, someone may say that God loves and that love is an emotion. But with respect to love, two points must be made. First, God’s love is eternal, therefore not a sudden change, therefore not an emotion. Second, God commands us to love him. A command requires voluntary obedience. Therefore the love God commands is volitional, not emotional. Doubtless God commands the impossible. He commands us to keep his law perfectly. This we cannot do because of sin. The impossibility arises from us; it does not arise from any irrationality in the commands. God commands the impossible, but he does not command the absurd. (Gordon Clark, Today's Evangelism: Counterfeit or Genuine?)A few scattered thoughts:
1. Given that emotions are involuntary, I don't think emotions can be intrinsically sinful, although their presence in a particular context could serve as an indication of sin. Jesus was sinless yet, at times, angry, happy, and sad. If we suppose someone else was happy when Jesus would have been angry (with money changers) or sad (at Lazarus' death), or if we suppose someone else was angry or sad when Jesus was happy, then it would seem emotions are a kind of contextually indicative reflection of one's current state of sanctification. Either way, any argument for the impassibility of God couldn't proceed along these lines.
2. If a "passion or affection is the result of being affected by some external force," then God could not be necessarily be affected by anything external to the Trinity, for there is nothing external to the Trinity which was not created. So if divine passibility could be true, it would have to be a contingent property like divine creator-ship.
[I am assuming here that the relationships among the members of the Trinity are necessary, eternal, and, thus, not of the sort intended to be included by a doctrine of divine impassibility. The Father, Son, and Spirit are distinct, eternal generation and spiration are true, and so some might be tempted to say that the members of the Trinity "affect" one another, for example - whether in respect of person or merely personhood is a matter of dispute between eastern and western Trinitarian models - but I would assume that necessary condition for passibility is change, which obviously does not take place given that the Trinity and their relationships are eternal, at least apart from consideration of a contingent creation (i.e. the ontological vs. economic Trinity)].
3. I believe creation is contingent so as to avoid, roughly, the consequence that the Trinity depends on creation in the same sense the Father depends on the Son. However, I think that most proponents of divine impassibility do not think that the fact that we can affirm contingent truths about God like "God is our creator," "God is our savior," etc. would count as a mark against divine impassibility - they may not even count as a mark against divine immutability. At the very least, those contingencies are true in virtue of God's own initiative.
Furthermore, as a Calvinist, I believe God determines all things. God initiates and determines changes, so only in a proximate sense could I say that something external to God has affected Him (not that I would say that). It may affect Him, but only insofar as God had taken the initiative in determining to create something external to Him for that express purpose. So one question is whether God created with a purpose, in part, that His creation would affect Him - change Him, in some way. It may be that God did not create with this intention and so did not determine that creation would affect Him - obviously, this would be in favor of divine impassibility.
But suppose He did create and determine all things with that intention. Another question is, given that any alleged affectation would ultimately derive from God's own initiative and determination, whether that alleged affectation can really by ascribed to something external to God, or if it is really God who is changing Himself. The latter might be the more accurate expression of the state of affairs. This could constitute an argument for divine impassibility, but it would be controversial.
Suppose I intend and determine to hit myself with a hammer and that nothing could stop me from doing so. Was the hammer instrumental to the change in pain I feel? Yes. But the hammer can has no instrumentality apart from my will. So was it the hammer that really affected me, or was it my will, or was it both? If the first or last option, that cuts against divine impassibility. If the second, that works in its favor. An answer to this could depend on the semantics and technical nuances of a particular expression of divine impassability. I'm not sure.
Of course, this all assumes the speculative point about whether God intended that creation would lead to some kind of internal change or affectation in the first place. But to my mind, the Incarnation seems to correspond to the hammer illustration: who and what I am allows me to be able to assume a hammer in my hand by which I can hurt myself. Indeed, the Incarnation is even stronger than this analogy, for whereas I cannot in fact create a hammer, the assumption of a hammer is not an assumption of a nature or capacities, and other things can prevent me from hurting myself with it, the Son in fact initiated and infallibly determined His own Incarnation and assumption of a nature or capacities precisely so as to be hurt through the crucifixion. So the questions about whether I am being really affected by the hammer, if I am affecting myself, or both would have analogous correspondents to the situation of the Incarnation.
4. Direct arguments for divine impassibility could be made from other alleged divine attributes. Divine simplicity would entail divine impassibility. Given that there would be no real distinction between impassibility and simplicity, then it is trivially true that any argument for divine simplicity would be an argument for divine impassibility. I have numerous reasons for disagreeing with divine simplicity, however, so to the extent that I agree with divine impassibility, I wouldn't look here.
Another direct argument for divine impassibility is to argue for divine immutability. Divine impassibility may not require divine immutability per point 3 above, but if God doesn't change, then obviously nothing external to Him could be said to change Him. Ultimately, what grounds our belief in something about God ought to be divine revelation, but our support can be either explicit or implicit. I'm sure others are more aware of relevant biblical passages than I am.
Currently, I am sympathetic towards divine immutability and, to that extent, sympathetic towards divine impassibility as well. I am undecided, however. Obviously, an adherent of divine immutability must consider the relevancy of the Incarnation. Unless I were to encounter incontrovertible, unambiguous biblical support for a specific theory and expression of immutability or impassibility - I may have done so and just don't remember, or I may have and just think (correctly or not) that those texts are prima facie consistent with varying positions - I would need to develop my views on the Incarnation and a theory of time before I could answer decisively.