In a discussion about its biography of Stalin, historian Stephen Kotkin start discussing Lenin's testament. And though he doesn't outright say it, he seems to have very strong doubts about the veracity of the document, mentionning that there is no original version, that the document comes at a very convenient time for Stalin's opponents, and that Lenin was already very sick at that time (and therefore, that it's doubtful whether he could have written that).
Is there a consensus on this topic ? I could not find more information on the matter, and the wikipedia page doesn't seem to question the veracity of the document.
Some relevant quotes I thought might be of interest:
"Another strange thing: of all those mentioned in the letter Stalin appears in the most favorable light. He is the one Lenin accuses of rudeness and intolerance, but that was never regarded as a fault in the proletarian party. " Radzinsky, Edvard. Stalin. New York: Doubleday, c1996, p. 208
"Footnote: Trotsky himself at first admitted that Lenin had left no Testament or Will. In a letter to the New York Daily Worker on August 8, 1925, Trotsky wrote: “As for the “will’, Lenin never left one, and the very nature of his relations with the Party as well as the nature of the Party itself made such a “will’ absolutely impossible. “In the guise of a “will’ the emigre and foreign bourgeois and Menshevik press have all along been quoting one of Lenin’s letters (completely mutilated) which contains a number of advices on questions of organization. “All talk about a secreted or infringed “will’ is so much mischievous invention directed against the real will of Lenin and of the interests of the Party created by him.” " Sayers and Kahn. The Great Conspiracy. Boston: Little, Brown and Company, 1946, p. 200
"Right from the first session of the Central Committee, after the 13th Congress, I asked to be released from the obligations of the General Secretaryship. The Congress itself examined the question. Each delegation examined the question, and every delegation, including Trotsky, Kamenev and Zinoviev, voted unanimously in favor of Stalin remaining at his post. What could I do then? Abandon my post? Such a thing is not in my character…. At the end of one year I again asked to be set free and I was again forced to remain at my post. What could I do then? " Stalin, Joseph. Stalin’s Kampf. New York: Howell, Soskin & Company, c1940, p. 244
I'm of the opinion that it almost certainly wasn't a fake. However it was not supposed to be published and it was not an "Order" to the party. It was just Lenin giving some advice after being quite ill and in a particularly bad mood.