Presumably you are referring to the idea advanced by the late Imam Khomeini, in which he argued that political, religious, and social affairs of the state should be administered by an Islamic jurist, and which has guided the approach to governance in Iran since the 1979 Revolution.
Is this supported by narrations? Advocates will say yes; opponents will say no.
I would say this is not literally specified by narrations, but neither are many ideas which we have have today. Rather, there is a sort of derivation and rational interpretation involved.
To review the arguments for this form of wilayat al-faqih (wilayat al-faqih al-mutlaqa, or an absolute authority of the jurist), I would recommend reading Chapter 11 of The Most Learned of the Shi'a: The Institution of the Marja' Taqlid, entitled "Analysis of Khomeini's Proofs for al-Wilaya al-Mutalqa (Comprehensive Authority) of the Jurist". (This book is easily available online in PDF.)
Imam Khomeini's arguments for this have also been also translated into English under the title Islamic Governance.
After you have read these things, you can make your own decision about whether or not they are supported by narrations, and also get a sense of the sociopolitical context which brought about these ideas.
If I may indulge in my opinion (keeping in mind that most everyone has an opinion, and my personal opinion was not solicited), I do not believe that Allah requires Muslims to be governed by or under the supervision of ulema, even if Islam offers some teachings about how a state should run. I am not of the view that Allah requires any specific type of political system; rather, I think that Muslims can choose any type of political system for themselves, and maybe there will be other political systems in the future that we can choose from. This is of course before the time of the Mahdi (A), at which time we follow him and his guidance.
However, I understand the sociopolitical context which led Muslims, in general, to put a lot of hope into the concept of the "Islamic state" in the 20th century, and the situation in Iran in particular (including the legacy of European colonialism) which led to a desire for this arrangement and the hope that it would lead to a juster and more prosperous society. It is fair to say that while Iran is an independent functional state which has done quite well given the immense challenges it has endured (including war, isolation, and sanctions), there have been some bumps in the road with respect to the way wilayat al-faqih has been enacted there, and it has not created the utopia that was hoped for. It seems that combining religion with the model of a secular nation-state and making it into a political ideology does not work well. However, to be fully fair in judging the concept, one would have to look at it in a variety of time periods (say, in different centuries past and future) to truly understand how the system works in a variety of world situations and challenges, not just in the face of the snapshot of the world today. We humans tend to be blindsighted and assume that the world we have around us is the gold standard for all time, but our circumstances today are quite unique in human history (as well as socially and environmentally unsustainable), so we don't know what the future will call for. On the flip side, outside Iran, in places where Shi'a tend to be disenfranchised, the authority of the ulema/maraji' tends to provide security and social stability in a way that the government does not, even if it is not formally recognized by the secular state.
As for wilayat al-faqih in a general (non-absolute sense), this just refers to the idea that the Islamic jurist has some forms of authority (for instance, judging between disputes, or being a guardian of certain people in some circumstances) and this is commonly accepted among Shi'a ulema.
