First off, Lenin was never removed from the official iconography of the Soviet Union upon Stalin's assumption of power. In fact, quite the opposite in that Lenin's image became ubiquitous in the Soviet Union. Walter Benjamin writing in 1927 noted that the veneration of Lenin after his death took on all the appearances of a cult with Lenin as a quasi-religious figure. As the Stalin's cult of personality grew in the USSR, the two figures operated in tandem.

That being said, the relationship between Lenin and Stalin in the 1920s was far from a platonic ideal. Lenin valued Stalin's drive and organizational skills in setting up the Party apparatus, especially since Lenin was debilitated because of his strokes and the aftereffects of an assassination attempt. Yet the personal relationship between the two men was not particularly close. Stalin also had a somewhat fraught relationship with Lenin's wife Nadezhda Krupskaya, especially as Stalin tried to interfere with her over issues on Lenin's health. Lenin had sent a note to Stalin rebuking Stalin over the latter's rough treatment of Krupskaya prior to his third stroke in 1923. But Lenin's apprehension over Stalin was not just because of his wife; Stalin was known as the "grey blur" in his role as General Secretary and was adept at setting up bureaucratic entities and managing them. This was one of Lenin's chief fears, namely, that the revolutionary energy of Marxism would be sapped by bureaucracy.

The issue of how close Lenin was to Stalin came to a head after the Lenin's death and the revelation of Lenin's last testament. In this document, likely written with the aide of Krupskaya, Lenin warned that Stalin has too much power as General Secretary and is quite coarse and dismissive in his dealings with his fellow Bolsheviks. The testament suggested that Stalin be removed from his position as General Secretary because he focused more on bureaucracy at the expense of the Bolsheviks' ideological mission.

Subsequent commentators, most prominently Trotsky, have looked at this suggestion of Lenin as a prophetic sign that Lenin did not desire Stalin in a position of power and that, by extension, the subsequent Stalinist dictatorship was an departure from the vision of Soviet power Lenin desired. Stalin himself was quite distraught over the testament, allegedly exclaiming that Lenin "shit on himself and then on us!" after he first read it. However, Lenin's testament was quite catty about pretty much the entire leadership cadre of the Soviet Union. And some of these charges were more ideologically severe than being rude or having too much power. The testament contends that Trotsky, who rather arrogantly considered himself Lenin's heir apparent, was not a Bolshevik prior to the 1917 Revolution. Bukharin's theories, according to Lenin "can be classified as fully Marxist only with great reserve, for there is something scholastic about him (he has never made a study of dialectics, and, I think, never fully understood it)." Lenin also contended that Zinoviev and Kamenev both made serious tactical mistakes during 1917. In short, no one came across well in Lenin's testament, but Stalin comes off slightly better than his contemporaries- Lenin only warned about Stalin's potential to abuse his position as General Secretary, which is a different matter than ideological nonconformity.

Although the testament circulated amongst the upper echelons of the Party, no one in the immediate post-Lenin leadership wanted the testament to be disseminated to the wider Soviet public and the world. Trotsky's purging and exile led to him publishing the testament in a pamphlet in 1934.

Now as to the question whether or not a magically-recuperated Lenin would have followed the same course as Stalin is an unanswerable question. It has become something of an article of faith among Trotskyites and some on the left that Stalin was a perversion of the ideals of 1917. This has given Lenin's testament an air of prophecy that it was a last exit for the Soviet experiment to escape from dictatorship. However, this interpretation ignores some of the salient problems that the Soviet Union faced that would have still existed without Stalin. The USSR was a proletarian state without much of a proletariat or heavy industry and the mixed economy of NEP did not do much to resolve this problem. The Civil War had imparted to the Bolshevik leadership a lesson about the efficacy of force and the decision-making process of the early Soviet Union favored a narrow party oligarchy calling the shots. I would argue that these preconditions made the USSR acutely conducive to a dictatorship. The Testament itself strongly implies that an individual's past mistakes or decisions are a criterion for removal from power.

Edit: tweaked a little for clarity.

Sources

Kotkin, Stephen. Stalin: Paradoxes of Power. New York: Penguin 2014.

Plamper, Jan. The Stalin Cult: A Study in the Alchemy of Power. Stanford: Yale University Press, 2012.

White, James D. Lenin: The Practice and Theory of Revolution. Houndmills, Basingstoke, Hampshire: Palgrave, 2001.

The last thing that Lenin ever wrote is known as his Testament. He planned to read it aloud during the 12th Congress of the Russian Communist party, but he was unable to do so due to his failing health. In it he outlines his plan for the future of the Soviet Union and endorses Trotsky over Stalin as his successor. From his Letter to Congress:

Comrade Stalin, having become Secretary-General, has unlimited authority concentrated in his hands, and I am not sure whether he will always be capable of using that authority with sufficient caution. Comrade Trotsky, on the other hand, as his struggle against the C.C. on the question of the People's Commissariat of Communications has already proved, is distinguished not only by outstanding ability. He is personally perhaps the most capable man in the present C.C., but he has displayed excessive self-assurance and shown excessive preoccupation with the purely administrative side of the work.

Lenin dies before he has a chance to voice his views publicly and by that time Stalin had pretty much assumed control. Lenin's widow Krupskaya pushed for it to be read aloud at the general assembly of the 13th congress but Stalin was able to limit its exposure.

After Lenin's death Stalin did his best to make it seem like he was best friend's with Lenin because it helped to legitimize his rule so maybe that's why some people think they were close, but I don't believe that was the case. The fact that he didn't want Stalin to succeed him shows Lenin didn't trust him at least.