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I will confine my answer to St. Thomas Aquinas' view, and I will divide it into two parts: first whether demons such as Satan do in fact love, and secondly whether they can love. All quotes are from Aquinas.
Perhaps other users can represent a non-Catholic or non-Thomistic view given the Meta discussion which promotes the practice of working on overview questions collaboratively (link).
Demons can be said to love in a natural way, but not in a supernatural way. First let us consider that demons do love in a natural way.
Spiritual beings, such as angels and demons, are possessed of an intellect and, more importantly, a will (ST Ia.59). The movement of the will is love:
Consequently, in the intellectual nature there is to be found a natural inclination coming from the will; [...] Therefore, since an angel is an intellectual nature, there must be a natural love in his will. (ST Ia.60.1.c)
...but in so far as [love and joy] express a simple act of the will, they are in the intellective part: in this sense to love is to wish well to anyone. (ST Ia.59.4.ad2)
Now the only sins of the demons are pride and envy (ST Ia.63.2), but both of these sins presuppose love of oneself. Envy occurs when "another's good is apprehended as one's own evil" (ST II.II.36.1.c), and the shunning of one's evil presupposes the love of one's good. "Consequently love must needs precede hatred; and nothing is hated, save through being contrary to a suitable thing which is loved" (ST II.I.29.2.c). Pride is an excessive "love of one's own excellence" (ST II.II.162.1.obi2). Thus all demons, including Satan, must at least love themselves.
Aquinas may even imply that demons love God insofar as God is the source of their natural being, but fail to love God in a supernatural way due to their mortal sin. He says,
It is natural for the angel to turn to God by the movement of love, according as God is the principle of his natural being. But for him to turn to God as the object of supernatural beatitude, comes of infused love, from which he could be turned away by sinning. (ST Ia.63.1.ad3)
This leads us to the sense in which demons do not love: by way of supernatural charity.
Demons do not love supernaturally. Thomas makes an argument that demons are Cherubim and not Seraphim because the demons sinned mortally and mortal sin is incompatible with the fiery charity of the Seraphim:
Cherubim is interpreted "fulness of knowledge," while "Seraphim" means "those who are on fire," or "who set on fire." Consequently Cherubim is derived from knowledge; which is compatible with mortal sin; but Seraphim is derived from the heat of charity, which is incompatible with mortal sin. Therefore the first angel who sinned is called, not a Seraph, but a Cherub. (ST Ia.63.7.ad1)
Indeed, no creature in a state of mortal sin possesses the supernatural love of charity. Since the demons are in a state of mortal sin they do not possess this kind of love. Indeed, it could even be said that in committing the sin of pride Satan loved himself more than he loved God. Thomas quotes Augustine, "[Satan] wished to enjoy his own power rather than God's" (ST II.II.163.2.c).
Thomas is clear that demons cannot change their minds and convert. "According to Catholic Faith, it must be held firmly both that the will of the good angels is confirmed in good, and that the will of the demons is obstinate in evil" (ST Ia.64.2.c). This is because the angelic intellect operates in such a way that once it has adhered to a thing, it does so immovably. So demons could have chosen to accept God's grace and enter into supernatural love, but having once decided to shun God, they can never go back again:
Therefore, if [the angel's] will be considered before its adhesion, it can freely adhere either to this or to its opposite (namely, in such things as he does not will naturally); but after he has once adhered, he clings immovably. So it is customary to say that man's free-will is flexible to the opposite both before and after choice; but the angel's free-will is flexible either opposite before the choice, but not after. (ST Ia.64.2.c)
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For a Catholic perspective, Pope Francis, who talks about Satan as much as he talks about anything else says that Satan does not know how to love.
The second attitude is that of Herodias, the wife of the king’s brother. “The Gospel says that she ‘hated’ John” because he spoke with clarity. And, the Holy Father continued, “we know that hatred is capable of anything”. Indeed, “hatred is the breath of Satan”, who does not know how to love”. Herodias “had the satanic spirit of hatred”.
If you don't know how to do a thing, it doesn't mean you're not capable, it just means you're not apt to do it.
A reformed perspective is more pragmatic, playing a little loose with the word love:
It makes sense that Luther would have such a strong experiential knowledge of the devil, for Satan loves nothing more than to attack the gospel, and Luther was at the forefront of the greatest recovery of the gospel
So Satan, if he does love, he loves doing evil. The meaning of the word love in that case would be about the basest possible meaning of love that I don't even know how it qualifies.
I don't believe that the Orthodox have a particularly extensive demonology department, which is probably to their credit! It might be hard to find an answer there.
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We hear from the mouth of Jesus himself :
How can Satan drive out Satan? If a kingdom is divided against itself, that kingdom cannot stand. If a house is divided against itself, that house cannot stand. - (Mk 3:23-25; Mtt 12:26; Lk 11:18).
The Lord is loud and clear in stating that Satan and his followers are united, whatever their intentions be. There cannot be unity in the group without an element of affinity among the members. If one could call that affinity “love”, Satan is capable of expressing love.
As for whether he is capable of receiving love, a reading of the Book of Job should show us that Satan was not in such bad terms with God in the beginning. See Job 1:6-12:
One day the heavenly beings came to present themselves before the Lord, and Satan also came among them. The Lord said to Satan, “Where have you come from?” Satan answered the Lord, “From going to and fro on the earth, and from walking up and down on it.” The Lord said to Satan, “Have you considered my servant Job? There is no one like him on the earth, a blameless and upright man who fears God and turns away from evil.” Then Satan answered the Lord, “Does Job fear God for nothing? Have you not put a fence around him and his house and all that he has, on every side? You have blessed the work of his hands, and his possessions have increased in the land. But stretch out your hand now, and touch all that he has, and he will curse you to your face.” The Lord said to Satan, “Very well, all that he has is in your power; only do not stretch out your hand against him!” So Satan went out from the presence of the Lord.
See the lightheartedness of the dialogue between God and Satan, especially in the opening sentences! What else is required to prove that Satan had his share of love, even from God?