Objection 1. It would seem that fear does not remain in heaven. For it is written (Proverbs 1:33): "He... shall enjoy abundance, without fear of evils," which is to be understood as referring to those who already enjoy wisdom in everlasting happiness. Now every fear is about some evil, since evil is the object of fear, as stated above (A2,5; I-II, 42, 1). Therefore there will be no fear in heaven.
Objection 2. Further, in heaven men will be conformed to God, according to 1 John 3:2, "When He shall appear, we shall be like to Him." But God fears nothing. Therefore, in heaven, men will have no fear.
Objection 3. Further, hope is more perfect than fear, since hope regards good, and fear, evil. Now hope will not be in heaven. Therefore neither will there be fear in heaven.
On the contrary, It is written (Psalm 18:10): "The fear of the Lord is holy, enduring for ever and ever."
I answer that, Servile fear, or fear of punishment, will by no means be in heaven, since such a fear is excluded by the security which is essential to everlasting happiness, as stated above (I-II, 5, 4).
But regard to filial fear, as it increases with the increase of charity, so is it perfected when charity is made perfect; hence, in heaven, it will not have quite the same act as it has now.
In order to make this clear, we must observe that the proper object of fear is a possible evil, just as the proper object of hope is a possible good: and since the movement of fear is like one of avoidance, fear implies avoidance of a possible arduous evil, for little evils inspire no fear. Now as a thing's good consists in its staying in its own order, so a thing's evil consists in forsaking its order. Again, the order of a rational creature is that it should be under God and above other creatures. Hence, just as it is an evil for a rational creature to submit, by love, to a lower creature, so too is it an evil for it, if it submit not to God, by presumptuously revolt against Him or contemn Him. Now this evil is possible to a rational creature considered as to its nature on account of the natural flexibility of the free-will; whereas in the blessed, it becomes impossible, by reason of the perfection of glory. Therefore the avoidance of this evil that consists in non-subjection to God, and is possible to nature, but impossible in the state of bliss, will be in heaven; while in this life there is avoidance of this evil as of something altogether possible. Hence Gregory, expounding the words of Job (26:11), "The pillars of heaven tremble, and dread at His beck," says (Moral. xvii, 29): "The heavenly powers that gaze on Him without ceasing, tremble while contemplating: but their awe, lest it should be of a penal nature, is one not of fear but of wonder," because, to wit, they wonder at God's supereminence and incomprehensibility. Augustine also (De Civ. Dei xiv, 9) in this sense, admits fear in heaven, although he leaves the question doubtful. "If," he says, "this chaste fear that endureth for ever and ever is to be in the future life, it will not be a fear that is afraid of an evil which might possibly occur, but a fear that holds fast to a good which we cannot lose. For when we love the good which we have acquired, with an unchangeable love, without doubt, if it is allowable to say so, our fear is sure of avoiding evil. Because chaste fear denotes a will that cannot consent to sin, and whereby we avoid sin without trembling lest, in our weakness, we fall, and possess ourselves in the tranquillity born of charity. Else, if no kind of fear is possible there, perhaps fear is said to endure for ever and ever, because that which fear will lead us to, will be everlasting."
Reply to Objection 1. The passage quoted excludes from the blessed, the fear that denotes solicitude, and anxiety about evil, but not the fear which is accompanied by security.
Reply to Objection 2. As Dionysius says (Div. Nom. ix) "the same things are both like and unlike God. They are like by reason of a variable imitation of the Inimitable"--that is, because, so far as they can, they imitate God Who cannot be imitated perfectly--"they are unlike because they are the effects of a Cause of Whom they fall short infinitely and immeasurably." Hence, if there be no fear in God (since there is none above Him to whom He may be subject) it does not follow that there is none in the blessed, whose happiness consists in perfect subjection to God.
Reply to Objection 3. Hope implies a certain defect, namely the futurity of happiness, which ceases when happiness is present: whereas fear implies a natural defect in a creature, in so far as it is infinitely distant from God, and this defect will remain even in heaven. Hence fear will not be cast out altogether.
Source: Summa Theologica (New Advent)