Added 4.14.16; updated 5.17.23 (original conversation 4.03.16)
Does Islam say love those who hate you? Or is that only in Christianity?
The example of the Prophet Muhammad ﷺ was to patiently call everyone to good and success, even those who did hateful or sinful things. He compared himself to someone catching grasshoppers to prevent them jumping in the fire. And if one of them did jump in the fire, meaning someone who had wronged him and rejected Islam passed away, he would pray for Allah's Mercy on them or that their children would be guided, if not them.
And his tolerance of harm from others was a means of attraction to Islam. A man used to leave garbage at the door of the Prophet ﷺ every day. The Prophet remained patient and did nothing about this until one day when there was no garbage. On that day, the Prophet ﷺ went to visit the man and ask if he were not feeling well. The man was surprised that while he had put garbage at the door of the Prophet, he returned that bad treatment with kindness. The man was impressed to the point that he embraced Islam as a new Muslim.
In another case, the Jewish scholar Zaid ibn Su`nah openly insulted the Prophet ﷺ and rudely demanded repayment of a loan before its agreed time. Umar was angered by this and told Zaid, "If I were not concerned with [the Prophet's] leaving us, I would have struck your head with my sword." The Messenger of Allah, who meanwhile had been keeping a calm demeanor, said, "O `Umar! We don't need this. I was more in need of your advice to pay off his loan well, and he your advice to deal courteously. Go with him, `Umar, pay off his loan, and give him twenty extra saa` (44 kilograms) of dates because you frightened him" (Transmitted by Tabarani, al-Mu`jam al-Kabeer, to which al-Haithami says that all of the narrators of the hadith are sound. Also transmitted by Ibn Majah, Ibn Hibban, and Hakim, among others).
There are many other examples like that from the lives of the Prophet ﷺ and righteous Muslims who demonstrated patience and good character in the face of poor treatment from others. As well, Islam especially implores the Muslim to treat family members kindly, even those who do not treat us well. Unfortunately, not all Muslims live up to that model, but it is part of the message of Islam and often a means by which people either drop hostility or even accept the religion. Kindness, by Allah, can soften people's hearts and draw out the best in them. And, regardless how others respond, the calm ones save themselves from useless arguments and headaches and earn reward from Allah.
Perhaps, though, I'm talking more about gentleness and forbearance than love, but that's a less ambiguous thing and it provides opportunities for all good things to grow, including love. From Abu Hurayrah (radhiallahu 'anhu) who said that Allah's Messenger (sallallahu 'alayhi wa sallam) said: "Indeed Allah is gentle and loves gentleness, and gives due to gentleness that which He does not give to harshness" (reported by Ibn Majah no. 3688 and Ibn Hibban no. 549).
As for love, we love what Allah loves and we despise what Allah despises, and with wisdom and patience that accords those motives constructive relationships where we care about each other. Ameen. The Prophet ﷺ said as prophets said before him, that none has faith until he loves for his brother what he loves for himself.