Does Maimonides ever refer to the Allegory of the Cave from Plato's The Republic?
The Rambam (Maimonides) writes in The Guide for the Perplexed Part II Chapter VI:
I wonder at the expression "contemplating", which is the very expression used by Plato. God, as it were, "contemplates the world of ideals, and thus produces the existing beings."
To me this sounds like the same idea expressed in the Allegory only from a different perspective. Looking at footnotes to this passage may lead you to find a direct reference if such exists.
While not a direct reference to Plato's cave, Rambam expresses an idea that is somewhat similar in his discussion of Jacob's ladder:
This ladder all may climb up who wish to do so, and they must ultimately attain to a knowledge of Him who is above the summit of the ladder, because He remains upon it permanently. It must be well understood that the term "upon it" is employed by me in harmony with this metaphor. "Angels of God" who were going up represent the prophets. That the term "angel" was applied to prophets may clearly be seen in the following passages: "He sent an angel" (Num. xx. 16); "And an angel of the Lord came up from Gilgal to Bochim" (Judges ii. 1). How suggestive, too, is the expression "ascending and descending on it"! The ascent is mentioned before the descent, inasmuch as the "ascending" and arriving at a certain height of the ladder precedes the "descending," i.e., the application of the knowledge acquired in the ascent for the training and instruction of mankind. This application is termed "descent," in accordance with our explanation of the term yarad (chapter x.).
This is similar to one of the themes of the cave, wherein the philosopher escapes from the cave and acquires knowledge and then must return to the cave to impart his knowledge to the other prisoners.
I'm not sure why no one mentioned this yet, but the Rambam uses imagery very similar to the Platonic cave at the very beginning of his Guide (Introduction, p. 7-8, Pines edition), when discussing the degrees of wisdom that separate people:
You should not think that these great secrets are fully and completely known to anyone among us. They are not. But sometimes truth flashes out to us so that we think that it is day, and then matter and habit in their various forms conceal it so that we find ourselves again in an obscure night, almost as we were at first. We are like someone in a very dark night over whom lightning flashes time and time again. Among us there is one for whom the lightning flashes time and time again, so that he is always, as it were, in unceasing light. Thus night appears to him as day. That is the degree of the great one among the prophets, to whom it was said: But as for thee, stand thou here by Me, and of whom it was said: that the skin of his face sent forth beams, and so on. Among them there is one to whom the lightning flashes only once in the whole of his night; that is the rank of those of whom it is said: they prophesied, but they did so no more. There are others between whose lightning flashes there are greater or shorter intervals. Thereafter comes he who does not attain a degree in which his darkness is illumined by any lightning flash. It is illumined, however, by a polished body or something of that kind, stones or something else that give light in the darkness of the night. And even this small light that shines over us is not always there, but flashes and is hidden again, as if it were the flaming sword which turned every way. It is in accord with these states that the degrees of the perfect vary. As for those who never even once see a light, but grope about in their night, of them it is said: They know not, neither do they understand; They go about in darkness. The truth, in spite of the strength of its manifestation, is entirely hidden from them, as is said of them: And now men see not the light which is bright in the skies. They are the vulgar among the people. There is then no occasion to mention them here in this Treatise.
Like the distinctions of light in Plato's allegory (those enchained who only see shadows off the fire vs. the escapee who eventually sees the light of the sun), we have here distinctions of people. Instead of a cave, we are all in the darkness of night. Like Plato, the Rambam sets the philosopher at the rarest, loftiest level, the one who can see the kind of light that emanates from lightning (i.e., knows truth, understands ultimate reality). Most of us, however, either see no light, or, in rare intervals, only the kind of light displaced by degree (refracted from stones).