Karl Popper may have been extremely critical of The Republic, but in order to do so he first took the position that The Republic is reflecting Plato's own philosophical views, and not those of Socrates. That might be a valid approach to the Socratic problem, but the separation conveniently frees Popper of having to deal with the fact that the majority of the Socratic corpus and Socrates' life aren't exactly compatible with the ideals of a philosopher king.
The Republic is an incredibly complex work of philosophy1, and thus quite open to interpretation. Naive readings of are certainly attractive to people or groups of people who see themselves as virtuous guardians, and even more attractive to people actively looking for renowned philosophical works with (even superficial) parallels to their own ideology, to base their propaganda upon.
In the Republic, Plato presents five different regimes, in order of preference, or more accurately in order of degeneration:
- Aristocracy, which the Republic advocates for,
- Democracy, and
Ignoring (for the moment) the fact that tyranny is at the end, Plato's aristocracy is technically a meritocracy, with some novel ideas for the era. For example Plato goes against the norm of hereditary citizenship and proposes that there's nothing that impedes a child to belong to a different caste than its parents. On the other hand, it's understandable how the fact that the middle caste's (the soldiers) main responsibility is to force the will of the philosopher kings to the majority would be appealing to despotic regimes.
Plato's warnings against tyranny though seem to be wilfully ignored by, well, every dictator that came after him. Plato argues that the tyrannical man is the worst form of man, and eventually will lose his power and his life will be in constant danger. Even if the tyrannical man is not killed, in revenge for his crimes, his fear will eventually consume him, to the point that he will end up self-imprisoned, afraid to leave his own house.
The fact that the Platonic Academy produced a number of tyrants, many of who had direct contact with Plato2, should also be taken into consideration as a quality that a despotic figure would possibly find attractive.
(all) That said, and unless a despotic figure directly acknowledges Plato as an influence, how are we supposed to know? Hitler's fascination with ancient Sparta, for example, can be deduced from his direct comparison of the German troops to the ancient Spartans, when he paraphrases the Epitaph of Simonides in Mein Kampf:
Traveller, when you come to Germany, tell the Homeland that we lie here, true to the Fatherland and faithful to our duty.
But even with a direct quote, how can we measure the significance of the influence? Especially when discussing people that in general tend to be on the... less mentally stable side.
In any case, and to finally answer your question, there's at least one instance of a despotic figure directly acknowledging Plato's influence, Ruhollah Khomeini, who studied Plato and Greek philosophy in general in the 1920s. Wikipedia cites his first book, Kashf al-Asrar (Uncovering of Secrets), as evidence of a direct acknowledgement of Plato's influence, but I couldn't find a freely available English translation. Next best thing I could find was Khomeini's NY Times obituary:
In the 1920's, Ayatollah Khomeini followed his tutor to Qum, where he completed his studies, worked as a teacher and became interested in Islamic mysticism and Plato's ''Republic,'' which may have helped shape his vision of an Islamic state led by a philosopher-king.
1 And perhaps even a musical work.
2 Even though Plato himself used the word "tyrant" pejoratively.