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.