The problem is there are complexities on both sides of the question and the more we get into details the complications increase.
Sufism is a vast heritage made of doctrines, practices, styles, figures and tariqas, with some shared and overlapping and with some other peculiar and unique.
But to keep things simple, if we take Sufism to be a focus on Islamic spirituality via canonical acts of worship and piety then there's no Muslim that can reasonably reject it, as this focus is firmly rooted in Quran and hadiths from both sects. Shia Islam can be in particular said to be Sufism par excellence in that Shia holy figures were known among other things for their deep piety and dedication to worship. We know tahajjud was one of the fixed features of the life of all Shia Imams and how much they valued having privacy with Allah.
If Sufism is about worshiping Allah via love, then that's again a familiar aspiration in Shia Islam as evidenced in the love-filled supplications by the Shia Imams as those in the Fifteen Supplications by Imam Sajjad (as) and many other accounts from the life of Shia Imams.
If Sufism is about allegiance to a Qutb or a spiritual master, then again that sounds like the Shia doctrine of Imama which consists among other things in accepting the Imams as spiritual guides to secrets of religion and gates to divine mercy.
Take the Sufi idea that a mystic must focus on cultivating the heart for Divine illumination and spiritual vision. This seem to be just the realized state in the Shia Imams who describe their hearts as treasuries of Divine secrets and instruments by which they witness Allah's grandeur and the Throne.
If Sufis talk about cosmic precedence of the light of the Prophet (ص) and sometimes that of Ali (ع) over the entire creation, then Shia hadith corpora offer even far more extensive and substantial hadiths stating the same thing with greater details and depth adding the remaining Shia Imams and the Holy Prophet's daughter Fatima (س) as parts of the eternal light from which both nubuwwa and wilaya emanated.
If Sufis consider Imam Ali (as) as a paragon of Islamic spirituality and piety, Shias share that belief in addition to the belief that this epitome of Islamic spirituality and wisdom must have also been the political successor to the Prophet (ص).
So given the above intriguing parallels, then where do the controversies come from?
It stems from some statements or doctrines by some famous Sufi figures that are perceived to clash with Islamic doctrine or Shia theology. These are some of the oft-cited issues:
- Sufis believe in wahdat al-wujud which is a heretical belief.
- Many Sufi figures praised Umar, Abu Bakr and Uthman.
- Ibn Arabi endorsed worship of the calf by the Israelites!
- Mawlawi uses obscene and sexually explicit language in his poems!
- Hafiz talks about wine, bodily attractions and young boys in his “mystical” poems!
As you see these are very specific objections that they raise which require separate discussions. But you can guess those Shia scholars who endorse the central Sufi doctrines have a more nuanced understanding and some of them actually reject all of these charges by arguing that, and I am representing them rather casually here:
- Wahdat al-wujud, far from implying pantheism, is the most consistent and deepest understanding of tawheed which is fully in agreement with Quran and hadiths
- Praise of caliphs by some famous Sufi figures might have been done under taqiyya or have been simply due to their lack of knowledge about the wrongdoings by caliphs as a result of the political biases that were prevalent in their times. Most Sufi figures in history were formally Sunni, after all, despite having intriguing similarities in some of their more esoteric beliefs to the Shias.
- Ibn Arabi is talking about ontological worship not legal worship, actual worship not intended worship! He means that even worshipers of calf actually worship Allah without realizing! Prophet Musa was reprimanding Harun in the Quranic story because Harun didn’t realize that it was by Divine Providence that Israelites ended up worshiping the calf!
- Sexually explicit language was a normal culture of the time and it was not exclusive to Mawlawi. Moreover, many words that today sound impolite had a polite import in those times. So accusations of indecency against Mawlawi are a result of anachronist interpretations of his poems.
- Hafiz's references to bodily pleasures are only metaphors for Divine love. Moreover, the same pleasures are found in Janna and they could have been actually intended as such by Hafiz. Hafiz's poems also make many references to Sufi theological and moral concepts and he is reputed to have been a Quran-memorizer. So it is unlikely that his references to bodily or sinful pleasures had been literal.
So as you see there's a lot of controversies when it comes to the details and particulars of some Sufi figures. So it's not a matter of all or nothing. When it comes to Sufis like Ghazzali there is little disagreement about the soundness of his beliefs and methods among most Muslims, apart from his properly Sunni beliefs for Shias. Go to Mawlawi, and there are some controversies due to his practice of Sufi dance and music. Go to Hafiz and many people including Shias find him to have been outright corrupt and decadent despite indications to the contrary. Go to Mansur Hallaj and again there are pros and cons. However, go to properly Shia mystic-scholars such Allama Tabataba'i or Ayatollah Khomeini and nobody would question their exceptional wisdom, uprightness and piety. And in the history of Shia fiqh there have been about a dozen respected scholars with Sufi tendencies that no Shia would find fault with. And like I said in the beginning, take the most basic and central Sufi doctrines such as worship and Light of Prophethood, and one can argue that Sufism is actually a diluted and/or misplaced form of Shia esoteric beliefs on Imama!