Some of the views expressed in this answer were first given in diverse comments. The comments were accordingly deleted after the posting of this answer.
As explained by Philip Klöcking in the comments, many versions of Nietzche book available online and in bookstores are versions controlled by Nietzsche's sister. Normally, since the 1970s, experts have been working on the original versions. Therefore, this answer which is based on two books on Nietzsche published in 2002 and 2018 should be valid.
Nietzsche was a proponent of violent despotism and was misogynistic
A response to the criticism that seeing Nietzsche as a proponent of misogynistic and violent despotism is a common misreading.
Recent accounts of Nietzsche managed to go through all the unfathomable Nietzsche's books, and found he was a proponent of violent despotism and misogyny. One of them was made by Ronald Beiner (Professor of Political Science at the University of Toronto), the other by Domenico Losurdo, an Italian historian, essayist, Marxist philosopher, and communist politician.
A passage from Ronald Beiner Dangerous Minds: Nietzsche, Heidegger, and the Return of the Far Right (2018) review, by marxandphilosophy.org.uk
Author Ronald Beiner connects Nietzsche’s affinities for feudalism with the philosopher’s critique of compassion, morality, and egalitarianism, and he shows how such despotism of thought was reproduced by the Nazi enthusiast Heidegger as well.
In Dangerous Minds, Beiner discusses the influence Nietzsche has had on notorious contemporary ultra-rightists such as the U.S.-based white supremacist Richard Spencer and the Russian neo-fascist Aleksandr Dugin, as well as the historical Italian fascist Julius Evola, who was an “explicit disciple of Nietzsche” (3). Like Evola, Spencer declares himself a Nietzschean, and Dugin swears by the iconoclast’s ominous statement that “man [sic] is something that should be overcome” (2, 12). These prominent figures of an increasingly powerful Fascist International find inspiration in Nietzsche’s aristocratic differentiation between the putatively “elect” and “unfit peoples” (4) as well as the philosopher’s anticipation of Nazism’s practice of große Politik (“great [or noble] politics”) in his militaristic critique of Otto von Bismarck from the right, as György Lukács points out in The Destruction of Reason (1952), and his “imperialistic critique of nationalism” (136n2). Today’s far-rightists also admire the Nazi Heidegger, who himself took a great deal from Nietzsche, particularly his critique of liberal modernity as nihilistic. To date, reports Beiner, Dugin has dedicated four volumes to discussing Heidegger, with “more to follow” (139n27).
A passage from Domenico Losurdo (2002) Nietzsche, the Aristocratic Rebel: Intellectual Biography and Critical Balance-Sheet review, by marxandphilosophy.org.uk
What singles out Losurdo’s book at this level is his unrelenting willingness to confront the darkest, most clearly reactionary strata within Nietzsche’s text: strata which he contends underlie the real, but more passing changes between his different phases, and reflect his continuing attempt to generate an intellectual “war machine” capable of uncovering the roots of, and thereby finally overcoming, contemporary progressivism in “two thousand years” of the slave revolt (28.1), looking back to the post-exilic Jewish prophets and Pauline Christianity (12.8, 15.2). These reactionary strata include Nietzsche’s early anti-Semitism or “Judaeophobia” in the years of the Wagner circle, and the later radicalization of his claims concerning the world-historical responsibility of the Jewish prophets and priests (“the most disastrous people in world history” (Nietzsche 1990, §24; 27.1; 27.3)) for inaugurating the slave revolt in morality (GM I, 8); his repeated claims concerning the timeless necessity of slavery for any higher culture (a truth which is like “the vulture that eats the liver of the Promethean promoter of culture” (Nietzsche 2009, 5; 1.12)); his unrelenting hostility to any forms of socialism, from the Paris Commune to the progressive stances of Christian and socialist anti-semites like Stöcker and Dühring in the 1880s; the profound misogyny and anti-feminism of his mature works; and most of all, the increasingly open denial of the “right to life”, and advocacy of eugenic proposals to sterilize (Nietzsche 1999b, 479; 1999a, 401–2; 19.3), control the marriages and births of (19.3; 20.1), and even instrument the “annihilation” (Vernichtung) of “millions of the malformed” (Nietzsche 1999d, 156; 19.1; 20.1; 24.4), “those who have turned out badly” (Schlechtweggekommenen), or entire “decadent races” (Nietzsche 1999c, 69, 547) in the name of breeding a new aristocracy which could reinstate rank order between human beings, and “avoid going to ruin at the sight of the suffering created thereby, the like of which has never been seen before” (Nietzsche 1999c, 98; 11.1; 19.1-19.5).
The origins of a Nietzsche-washing
Criticizing Nietzsche as a proponent of far-right ideology is often faced with irony or an advised sigh, and a short phrase which often means something like "your simple and puerile mind was not able to grasp the unlimited profoundness and sophistication of the Genius oeuvre".
What are the origins of this Nietzsche-washing, that prevents from understanding many of his own's writings (e.g. "Every enhancement of the type ‘man’ has so far been the work of an aristocratic society – and it will be so again and again – a society that believes in a long scale of orders of rank and differences in value between man and man, and that needs slavery in some sense or other.” Beyond Good and Evil) as what they mean?
The answer seems to be the 1960's postmodern and critical theory philosophers.
According to the same marxandphilosophy.org.uk article synthesizing Beiner's book, Beiner also saw that Nietzsche was a great source of inspiration to thinkers such as the Frankfurt school (Horkheimer, Adorno) and the French post-structuralists (Foucault, Guattari, Deleuze).
This is also the view of Losurdo:
Despite the harshness of Nietzsche’s language in these kinds of passages, left-Nietzscheans such as Gianni Vattimo and Gilles Deleuze have attempted to allegorise or metaphorise these radical concepts on life and their relation to the will to power and the eternal return. Losurdo reveals the absurdity of such an approach that would discount any historical-social origins to the theory and ignore the brutality and danger with which Nietzsche seeks to shock his readers. Hence, the usual interpretation of Nietzsche as a ‘life-affirming’ philosopher is brought to bear on a darker political implication by Losurdo’s rendering here, knowing that where Nietzsche says life, he also states ‘the great majority of men have no right to existence’ (Nietzsche 1967: 464).
Nietzsche and relativism
Concerning Joseph Weissman readings of Nietzsche: "In fact a key question here is — quid juris, who judges? On the one hand, who has experienced all the varieties of life such that they could assign it a stable value? On the other, what could be more valuable than the life we are given; what could be more lovely, to a healthy-minded sensibility, than precisely the life we have? So that we assign life at once an infinite value; and also recognize it is inestimable, since we would need to experience every form of life in order to render a judgment.".
Nietzsche said he was anti-nihilist, but here on the contrary it shows Nietzsche was a radical-relativist, in particular a nihilist. No common understanding of the world, no common principles according to live can be arrived at and agreed on, since understanding and values depend exclusively on each individual subjective and sensorial experience.
So this is a theory of despotism as exposed from the point of view of the despot: I impose on you and everybody my understanding and my principles because this is how I feel it and want it, and I don’t have to justify myself. Hitler & Cie did not think differently for sure.
Responses to frequent rebuttals on critics of Nietzsche
- You misinterpreted Nietzsche
"The worst readers are those who behave like plundering troops: they take away a few things they can use, dirty and confound the remainder, and revile the whole." Human, All Too Human
"I am frightened, by the thought of what unqualified and unsuitable people may invoke my authority one day. Yet that is the torment of every teacher … he knows that, given the circumstances and accidents, he can become a disaster as well as a blessing to mankind." (A letter by Nietzsche to his sister, 1884)
Most intellectuals are not misappropriated or misinterpreted. If it is the way it is for Nietzsche, at the least, he is responsible for it (he was not coherent and clear, nor efficient in conveying ideas), at the worst, it is for good reasons (he was a bipolar psychopath).
If people say bad things about him based on his self-contradictory obscure speech, it is because those people are stupid, not because he did not produce a coherent, original and likeable oeuvre. And in any way, it is not his reponsibility. What an example for generations of people and philosophers.
Anyway, like it or not, the far-right Loves Nietzsche.
- Ok Nietzsche was misinterpreted
Let's assume Nietzsche was misinterpreted. Still, the role of an intellectual is twofold: (1) to produce original and efficient ideas (2) to express his/her idea in a way that it reduces to the maximum the chance of being misinterpreted. So even from this assumed point of view, Nietzsche is a bad and dangerous intellectual.
- Nietzsche was an innocent brilliant genius, all is the fault of his sister and mother
Here is what Losurdo says about this, as synthesized by Rory Jeffs, from marxandphilosophy.org.uk:
Bearing on these sections of the book that dare to go into the eugenic question, the issue of the Nazi ‘appropriation’ is also inevitably addressed by Losurdo. He argues that the rehabilitative work of Nietzsche’s postwar editors (namely, Kaufmann and Colli and Montinari) was successful largely due to their attribution to Nietzsche’s sister, Elisabeth Förster-Nietzsche, as the key instigator in rendering a Nazi-friendly Nietzsche in her assemblage and ‘forgery’ of the posthumous editions of The Will to Power (1901-06). However, Losurdo argues such defences of Nietzsche discount several important historical details. Firstly, he claims the official account of Elisabeth’s role in creating Nietzsche’s anti-Semitism is an ‘unsustainable conspiracy theory’ (711-15). Nietzsche’s defenders on this front never address Elisabeth’s own distancing of Nietzsche from anti-Semitism in her biography of him (Förster-Nietzsche 1895-1904). Furthermore, there is never any discussion of the fact that Nietzsche was attracting a right-wing audience of his published works before The Will to Power was released (566, 720-22). Whilst this does not necessarily resolve the issue of Nietzsche’s influence on Nazism, it does reveal something arbitrary about the ‘hermeneutics of innocence’ when it comes to the distinctions it makes over the ideological precursors to the Third Reich.
- Nietzsche is the Genealogy of the Moral, and this book is great
No. Nietzsche is Nietzsche, an author who wrote around 30 books, and must be judged on his whole oeuvre and his personal biography.
- Nietzsche is useful, because he proposed a criticism of the herd mentality
Yes, and Nietzsche wrote a theory for slavery-based misogynistic aristocracy instead.
The protestant reformation was much much more efficient in liberating people under the catholic church rule than Nietzsche.
- Nietzsche is great because his guiding principle was to provoke people into developing their own views.
This is not called philosophy. Philosophy, as science in general, is to propose an original interpretation of facts. This interpretation can then be criticized and improved.
If this is not what Nietzsche did, ok, but then he was not a philosopher.
- Nietzsche was an artist-philosopher, "His ubermensch was above all creative force"
If for him being obscure and lyric is a way of promoting creativity, ok, go for it, but it is not philosophy, it is art and poetry. You can not say you are a philosophe when it pleases you, and then switch into saying you are a poet when people criticize what you literally said, it is just too easy.
I think it is a good technique used since the romantics (to which Nietzsche, IMO, very well belongs) to become famous and authoritative by telling obscure and ambiguous things (which often turn around violence, sex, and drama) in a grandiose manner to appear profound and original, while telling nothing really interesting underneath.
IMO, Nietzsche, when he said "God is dead", wanted to take over the influence (at least psychologically, culturally and spiritually) that the Church used to have before the Enlightenment thinkers took its power down. I think it was also the case of the romantics. That is why they all made great resort to emotions and allegory, had such fascination for the middle ages and mysticism, and such detestation toward the Enlightenment.
"Behold, I am a prophet of the lightning and a heavy drop from the cloud: but this lightning is called Übermensch." Thus Spoke Zarathustra, Prologue.