Matthew 5:43-47 (RSV) “You have heard that it was said, `You shall love your neighbor and hate your enemy.’  But I say to you, Love your enemies and pray for those who persecute you,  so that you may be sons of your Father who is in heaven; for he makes his sun rise on the evil and on the good, and sends rain on the just and on the unjust.  For if you love those who love you, what reward have you? Do not even the tax collectors do the same?  And if you salute only your brethren, what more are you doing than others? Do not even the Gentiles do the same?”
Luke 6:27-29 “But I say to you that hear, Love your enemies, do good to those who hate you,  bless those who curse you, pray for those who abuse you.  To him who strikes you on the cheek, offer the other also; and from him who takes away your coat do not withhold even your shirt.” (cf. 6:35)
Can we find this teaching of Jesus in the Old Testament? Or is it exclusively a New Testament innovation? First of all, Jesus Himself said that His teachings were a development of existing ones in the Old Testament:
Matthew 5:17-18 Think not that I have come to abolish the law and the prophets; I have come not to abolish them but to fulfil them. For truly, I say to you, till heaven and earth pass away, not an iota, not a dot, will pass from the law until all is accomplished.
Romans 3:31 Do we then overthrow the law by this faith? By no means! On the contrary, we uphold the law.
Psalm 139:19-22 O that thou wouldst slay the wicked, O God, and that men of blood would depart from me,  men who maliciously defy thee, who lift themselves up against thee for evil!  Do I not hate them that hate thee, O LORD? And do I not loathe them that rise up against thee?  I hate them with perfect hatred; I count them my enemies.
This looks “bad” at first glance, but upon close examination, I think it is simply expressing the thought that “those whom God judges, I also judge [in agreement with God’s prior decision]; they deserve their condemnation.” It’s expressing the notion of judgment or proper justice or righteous indignation at those who indeed are among the reprobate, on their way to hell. And of course, Jesus also condemns wickedness and is appointed by the Father to judge the world at the end of the age.
But that’s all I could find in the Old Testament. On the other hand, there are passages that come quite close to what Jesus said about love of enemies:
Proverbs 25:21 If your enemy is hungry, give him bread to eat; and if he is thirsty, give him water to drink;
Romans 12:16-21 Live in harmony with one another; do not be haughty, but associate with the lowly; never be conceited.  Repay no one evil for evil, but take thought for what is noble in the sight of all.  If possible, so far as it depends upon you, live peaceably with all.  Beloved, never avenge yourselves, but leave it to the wrath of God; for it is written, “Vengeance is mine, I will repay, says the Lord.”  No, “if your enemy is hungry, feed him; if he is thirsty, give him drink; for by so doing you will heap burning coals upon his head.”  Do not be overcome by evil, but overcome evil with good.
Here are more:
Exodus 23:4-5 “If you meet your enemy’s ox or his ass going astray, you shall bring it back to him.  If you see the ass of one who hates you lying under its burden, you shall refrain from leaving him with it, you shall help him to lift it up.”
Job 31:29 “If I have rejoiced at the ruin of him that hated me, or exulted when evil overtook him [in context, this is bad or condemned behavior]
Proverbs 24:17 Do not rejoice when your enemy falls, and let not your heart be glad when he stumbles;
The prophet Elisha counseled mercy towards Syrian prisoners of war. The result was that they no longer raised Israel:
2 Kings 6:21-23 When the king of Israel saw them he said to Eli’sha, “My father, shall I slay them? Shall I slay them?”  He answered, “You shall not slay them. Would you slay those whom you have taken captive with your sword and with your bow? Set bread and water before them, that they may eat and drink and go to their master.”  So he prepared for them a great feast; and when they had eaten and drunk, he sent them away, and they went to their master. And the Syrians came no more on raids into the land of Israel.
Romans 1:19-10 For what can be known about God is plain to them, because God has shown it to them.  Ever since the creation of the world his invisible nature, namely, his eternal power and deity, has been clearly perceived in the things that have been made. . . .*Romans 2:14-15 When Gentiles who have not the law do by nature what the law requires, they are a law to themselves, even though they do not have the law.  They show that what the law requires is written on their hearts, while their conscience also bears witness and their conflicting thoughts accuse or perhaps excuse them