Objection 1. It would seem that to tempt is not proper to the devil. For God is said to tempt, according to Genesis 22:1, "God tempted Abraham." Moreover man is tempted by the flesh and the world. Again, man is said to tempt God, and to tempt man. Therefore it is not proper to the devil to tempt.
Objection 2. Further, to tempt is a sign of ignorance. But the demons know what happens among men. Therefore the demons do not tempt.
Objection 3. Further, temptation is the road to sin. Now sin dwells in the will. Since therefore the demons cannot change man's will, as appears from what has been said above (Question 111, Article 2), it seems that it is not in their province to tempt.
On the contrary, It is written (1 Thessalonians 3:5): "Lest perhaps he that tempteth should have tempted you": to which the gloss adds, "that is, the devil, whose office it is to tempt."
I answer that, To tempt is, properly speaking, to make trial of something. Now we make trial of something in order to know something about it: hence the immediate end of every tempter is knowledge. But sometimes another end, either good or bad, is sought to be acquired through that knowledge; a good end, when, for instance, one desires to know of someone, what sort of a man he is as to knowledge, or virtue, with a view to his promotion; a bad end, when that knowledge is sought with the purpose of deceiving or ruining him.
From this we can gather how various beings are said to tempt in various ways. For man is said to tempt, sometimes indeed merely for the sake of knowing something; and for this reason it is a sin to tempt God; for man, being uncertain as it were, presumes to make an experiment of God's power. Sometimes too he tempts in order to help, sometimes in order to hurt. The devil, however, always tempts in order to hurt by urging man into sin. In this sense it is said to be his proper office to tempt: for thought at times man tempts thus, he does this as minister of the devil. God is said to tempt that He may know, in the same sense as that is said to know which makes others to know. Hence it is written (Deuteronomy 13:3): "The Lord your God trieth you, that it may appear whether you love him."
The flesh and the world are said to tempt as the instruments or matter of temptations; inasmuch as one can know what sort of man someone is, according as he follows or resists the desires of the flesh, and according as he despises worldly advantages and adversity: of which things the devil also makes use in tempting.
Thus the reply to the first objection is clear.
Reply to Objection 2. The demons know what happens outwardly among men; but the inward disposition of man God alone knows, Who is the "weigher of spirits" (Proverbs 16:2). It is this disposition that makes man more prone to one vice than to another: hence the devil tempts, in order to explore this inward disposition of man, so that he may tempt him to that vice to which he is most prone.
Reply to Objection 3. Although a demon cannot change the will, yet, as stated above (Question 111, Article 3), he can change the inferior powers of man, in a certain degree: by which powers, though the will cannot be forced, it can nevertheless be inclined.
Source: Summa Theologica (New Advent)