Unbelievers are described by Muhammad (in the Qur'an) as "the vilest of animals" and "losers." Christians and Jews are hated by Allah to the extent that they are destined for eternal doom as a result of their beliefs. It would make no sense for Muhammad to then recommend them to be taken in as friends by Muslims.
In fact, the Qur'an plainly commands believers not to take unbelievers as friends.
Sura (5:51) - "O you who believe! do not take the Jews and the Christians for friends; they are friends of each other; and whoever amongst you takes them for a friend, then surely he is one of them; surely Allah does not guide the unjust people."
Sura (5:80) - "You will see many of them befriending those who disbelieve; certainly evil is that which their souls have sent before for them, that Allah became displeased with them and in chastisement shall they abide." Those Muslims who befriend unbelievers will abide in hell.
Sura (3:28) - "Let not the believers Take for friends or helpers Unbelievers rather than believers: if any do that, in nothing will there be help from Allah..."
Sura (3:118) - "O you who believe! do not take for intimate friends from among others than your own people, they do not fall short of inflicting loss upon you; they love what distresses you; vehement hatred has already appeared from out of their mouths, and what their breasts conceal is greater still; indeed, We have made the communications clear to you, if you will understand." This verse not only warns Muslims not to take non-Muslims as friends, but it establishes the deep-seated paranoia that the rest of the world is out to get them.
Sura (9:23) - "O ye who believe! Choose not your fathers nor your brethren for friends if they take pleasure in disbelief rather than faith. Whoso of you taketh them for friends, such are wrong-doers" Even family members are not to be taken as friends if they do not accept Islam. (This is the mildest interpretation of this verse from the 9th Sura, which also advocates "slaying the unbeliever wherever ye find them").
Sura (53:29) - "Therefore shun those who turn away from Our Message and desire nothing but the life of this world."
Sura (3:85) - "And whoever desires a religion other than Islam, it shall not be accepted from him, and in the hereafter he shall be one of the losers."
Sura (3:10) - "(As for) those who disbelieve, surely neither their wealth nor their children shall avail them in the least against Allah, and these it is who are the fuel of the fire." Those who do not believe in Muhammad are but fuel for the fire of Hell (also 66:6, 2:24. 21:98).
Sura (7:44) - "The Companions of the Garden will call out to the Companions of the Fire: "We have indeed found the promises of our Lord to us true: Have you also found Your Lord's promises true?" They shall say, "Yes"; but a crier shall proclaim between them: "The curse of Allah is on the wrong-doers" Muslims in heaven will amuse themselves by looking down on non-Muslims in Hell and mocking them while they are being tortured (see 22:19-22.
Sura (1:5-7) - "Show us the straight path, The path of those whom Thou hast favoured; Not the (path) of those who earn Thine anger nor of those who go astray" This is a prayer that Muslims are supposed to repeat each day. "Those who earn Thine anger" specifically refers to Jews and "those who go astray" refers to Christians (see Bukhari (12:749)).
From the Hadith:
Muslim (1:417) - Taken to mean that one's own relatives should not be taken as friends if they are not Muslim.
Abu Dawud (41:4815) - "The Prophet (peace be upon him) said: A man follows the religion of his friend; so each one should consider whom he makes his friend."
Bukhari (59:572) - "O you who believe! Take not my enemies And your enemies as friends offering them (Your) love even though they have disbelieved in that Truth (i.e. Allah, Prophet Muhammad and this Quran) which has come to you."
Ishaq 262 - "Some Muslims remained friends with the Jews, so Allah sent down a Qur'an forbidding them to take Jews as friends. From their mouths hatred has already shown itself and what they conceal is worse"
Ishaq 252 - The story of a young man who converts to Islam after hearing Muhammad. He then tells his own father that he can no longer have anything to do with him because, "I have become a Muslim and follow the religion of Muhammad." (To maintain a relationship with his son, the father "converts" as well). This is an important passage because it establishes that the principle of shunning is based merely on the status of non-Muslims as unbelievers, not on their relations toward Muslims. In this case, the father desperately loved his son and meant him no harm.
Even though they are explicitly kufr (unbelievers, Sura 5:17, Sura 4:44-59) Jews and Christians are given special status in Islam. So, if Muhammad warned believers against taking them as friends, then it surely is not permissible for Muslims to befriend atheists or those of other religions.
Some Muslims interpret this to mean that they should not even act friendly toward nonbelievers. (Most, fortunately, do not).
Some Muslims are embarrassed by verse 5:51 and have gone to elaborate lengths to modify its intent by interpreting the word 'friend' as "guardian" or "protector" - which are just two of several legitimate translations of the Arabic word. According to these apologists, the verse is referring to a Muslim's allegiance to a non-Muslim government (which is not all that comforting either). This appears to be refuted by the verse itself, which distinguishes between friends and protectors and instructs Muslims to avoid both.
Other apologists point to (Sura 60:8-9) which says that Allah doesn't necessarily forbid showing kindness to unbelievers, but to shun the ones "who warred against you on account of religion and have driven you out from your homes and helped to drive you out." This is a definite reference to the Meccans, whose leaders expelled Muhammad and his handful of followers from Mecca (following his declaration of war against them). The verse was narrated shortly after their arrival in Medina, when is was necessary for the Muslims to build alliances with non-Muslims in order to survive. The verses quoted above from Suras 9 and 5 are given much later, when Muslims had power, and they expand the scope of unfriendliness to include those who are not Muslim.
Modern apologists such as Jamal Badawi try to cloud the straightforward interpretation of verse 5:51 by pointing out that Muslims are allowed to take non-Muslims as marriage partners, thus implying friendship. In fact, verse 2:221 explicitly forbids Muslims from marrying unbelievers, even though verse 5:5 allows it (Allah's change of mind corresponded somewhat curiously with Muhammad's own desire to marry a non-Muslim woman). Yusuf Ali reconciles the contradiction by saying that non-Muslims wives are "expected" to become Muslim.
In any event, only Muslim men are allowed to marry outside the faith. The women they marry relinquish control over their own lives, even to the extent that they cannot raise their own children in their own faith. All children must be raised Muslim. The non-Muslim woman also agrees to a lifetime of sexual servitude, and may be beaten if she does not submit.
This certainly doesn't sound like friendship to the rest of us. If your local Muslim cheerleader tries to pretend otherwise, then simply ask if a non-Muslim man may enter into this sort of "friendship" with a Muslim woman... then sit back and watch the backpedaling.
On the whole, Islam is very clear in teaching that there is no equality between believers and unbelievers, and hence no basis for a relationship of peers. Those who do not profess Muhammad are intended to exist in subjugation to those who do, then spend eternity in Hell. This does not preclude Muslims from acting friendly toward others, of course, but this does not constitute friendship as it is generally understood in the modern world.