What does Sheikh Saduq mean when he says friendship with unbelievers is only possible under Taqiyya? Allama Tabatabai also explains about the Taqiyya verses that friendship can have a psychological effect on Muslims, so perhaps it is referring to close friendship?

Quotation:
And Allah, the Blessed and Exalted, has described the showing of friendship to unbelievers as being (possible only) in the state of taqiya. And He the Mighty and Glorious says:

“Let not believers take disbelievers for their friends in preference to believers. Whoso doeth that hath no connection with Allah unless (it be) that ye but guard yourselves against them, for fear of being killed” (Qur'an 3:28).3

And Allah the Mighty and Glorious says:

“Allah doth not forbid you to deal with kindness and fairness toward those who have not made war upon you on account of your religion, or driven you forth from your homes: for Allah loveth those who act with fairness (Qur'an 60:8).”

http://www.shiavault.com/books/a-shi-ite-creed/chapters/42-concerning-dissimulation-taqiyya#fn3

Also the opinion of As-Sayid ar-Radi:

وقد علمنا ان التقية لا تدخل الا في الظاهر دون ما في الضمير الباطن – لان من خوف غيرهليفعل امرا من الامور اذا كان من افعال القلوب لا يتمكن من معرفة حقيقة ما في قلبه، وانمايستدل باظهار لسانه على ابطال جنانه – فالذي يحسن عند التقية اظهار مولاة الكفار قولابالخلاط والمقاربة وحسن المعاشرة والمخالطة ويكون القلب على ما كان من قبل اضمارعداوتهم واعتقاد البراءة منهم وينوي الانسان بما يظهره من ذلك مأربين الكلام واحتمالاتالخطاب.

" … and we know that taqiyyah affects only externally (i.e.' its effect is only on the tongue) and not internally (on the heart, spirit, soul). When someone forces another one to do something, (and that thing happens to be connected with heart) then the oppressor has no way of knowing that his wish has been complied with, except through some uttering’s by the tongue from which he will conclude that he has succeeded in changing the heart of the oppressed. Therefore, the best course of action at the time of taqiyyah is to show the friendship with the unbelievers by words, to mix with them and live with them with good manners, but the heart must remain firmly with the previous feelings towards them, of hidden enmity and the belief of aloofness from them.

http://www.shiavault.com/books/taqiyyah/chapters/1-taqiyyah

by

1- First of all, we have to see what the word اولياء means. Many scholars have noted that it doesn’t just mean “friends.” It means guardians, protectors, or very close confidants. Muslims are prohibited from giving non-Muslims the upper hand, from submitting to them, and taking them as guardians who make decisions for them. Yes, in times of Taqiyyah, they can pretend to be very close to their enemies so they don’t kill them.

2- We have to understand the socio-political circumstances of these verses. At the time, Muslims had many enemies who wanted to uproot Islam. Sometimes those enemies would pretend to be the friends of Muslims, fool and deceive them with their sweet words, and once they would earn the trust of Muslims they’d make a surprise attack, undermine them, or conspire against them. The Qur’an prohibited Muslims from falling to such deception. But if this is not the case, then it’s ok yo have a non-Muslim friend.

3- If having a non-Muslim friend (who is very close, like a best friend) puts one’s faith at risk (the non-Muslim friend influences you such that your faith is weakened), then it would be haram to have such a friend. But this also applies to Muslim friends as well. Having a corrupt Muslim friend who will weaken your faith is haram.