Do any denominations teach that Moses was talking with three distinct beings in the burning bush?

There is a problem with the question because it speaks of "distinct beings." Christianity generally rejects the idea of "three beings" in the Trinity.

The question's title is different from its explanatory text, where "three persons" are mentioned rather than three "beings." This is important because the Nicaean Creed as understood in the West is "one being in three persons."

The son is "consubstantial" with the Father, a translation of the Greek h*m*ousios meaning "same substance." It can also be translated as "same being."

The Council of Nicea chose the word, “h*m*-ousios,” or same-being, to describe the relationship between God the Father and God the Son. ref: Consubstantial with the Father by Fr. Bryan Babick, SL.L.

This idea remained dominant among the theologians of winning side in the debates over the nature of the Trinity. Thus, it is only one "being" in traditional Christianity, not three. It is outside of mainstream historical Christianity to speak of "three beings."

As to "three persons," the Burning Bush is a Jewish theophany. The basic prayer of Judaism, the Shema Israel, affirms that God is One, not three. It's true that the text refers to the "angel of the Lord" in the burning bush, but he is the messenger of God, who is One. So the attempt to understand this episode in trinitarian terms is anachronistic. Angel is not a personal name here, it is an office. The angel is the agent of the One God, who is called both Yahweh and Elohim in the various sources from which the story is drawn.

"Do any denominations teach that Moses was talking with three distinct beings?" Not to my knowledge. If there is indeed any such denomination it must be small and obscure, not to mention outside the theological norms of Christianity, which joins with Judaism in affirming the oneness of God's being.

The question makes some unwarranted assumptions, probably on account of lack of awareness of the meanings and grammar of the original Hebrew text, that then lead toward incorrect conclusions. The three "beings" listed as distinct are not necessarily distinct at all when one examines the Hebrew text.

"Jehovah" is clearly God's name; usually translated in the King James Version (KJV) as "the LORD." Of the three words associated with this question, Jehovah is the only one that is a name.

"Elohim" in Hebrew can be used in either a singular or a plural context. The word itself has broad application, and is not always translated as "God" or "god/gods" (there is no case distinction in Hebrew). Elohim can reference any "supernatural" (spirit) being, inclusive of the true God, false gods (e.g. idols), both heavenly and fallen angels, or even a superior being, such as a human judge.

The word "elohim" is generally applied to angels in Psalm 8:5.

For thou hast made him a little lower than the angels [elohim], and hast crowned him with glory and honour. (Psalm 8:5, KJV)

It is applied to human judges in the following verses.

Then his master shall bring him unto the judges [elohim]; he shall also bring him to the door, or unto the door post; and his master shall bore his ear through with an awl; and he shall serve him for ever. (Exodus 21:6, KJV)

⁸ If the thief be not found, then the master of the house shall be brought unto the judges [elohim], to see whether he have put his hand unto his neighbor's goods. ⁹For all manner of trespass, whether it be for ox, for ass, for sheep, for raiment, or for any manner of lost thing which another challengeth to be his, the cause of both parties shall come before the judges [elohim]; and whom the judges [elohim] shall condemn, he shall pay double unto his neighbor. (Exodus 22:8-9, KJV)

And to a human judge here in the KJV, whereas some versions might prefer to translate here as "God" (the term is somewhat ambiguous, making translation a little subjective):

If one man sin against another, the judge [elohim] shall judge him: but if a man sin against the LORD, who shall entreat for him? Notwithstanding they hearkened not unto the voice of their father, because the LORD would slay them. (1 Samuel 2:25, KJV)

In the story of King Saul seeking counsel of a spirit medium (one having "familiar spirits" per KJV), "elohim" applies to evil spirits, called gods (KJV):

And the king said unto her, Be not afraid: for what sawest thou? And the woman said unto Saul, I saw gods [elohim] ascending out of the earth. (1 Samuel 28:13, KJV)

"Elohim" is applied to Abraham (actually a noun in Hebrew, not an adjective as translated here in English).

Hear us, my lord: thou art a mighty [elohim] prince among us: in the choice of our sepulchres bury thy dead; none of us shall withhold from thee his sepulchre, but that thou mayest bury thy dead. (Genesis 23:6, KJV)

[Note: This particular text may have a better translation, such as that Abraham is called a "prince of God"--a proper translation if the Hebrew nouns are determined to be in construct chain form; however, "elohim" can apply to people, and has potentially broad application, so it is not necessarily incorrect to see Abraham addressed as "elohim."]

"The angel of the Lord" refers to the representative of Jehovah (Yahweh), most probably a reference to Christ in this context, but certainly an "elohim" regardless.

See more about the word "elohim" in Hebrew HERE.

Given a proper understanding of the Hebrew, it becomes clear that a conclusion of three "beings" here is untenable. Only one of the three terms considered is actually a name, one is used to mean a representative or messenger, and "elohim" applies to any superior or supernatural being, including God. All three terms might properly be applied to a single being.

To claim that there are three "names" is erroneous, and leads to an incorrect understanding.

If one were to say: "a delegate is coming, the secretary of the president" -- how many are coming? Even changing "president" to a specific name would not increase the number who were coming. All three terms would still address a single personage who was coming.

Conclusion

I am unaware of any denomination which teaches that this burning-bush passage in Exodus addresses three separate beings. If any did, however, it would but demonstrate the lack of Hebrew scholarship within that denomination, for the Hebrew cannot be used to establish the existence of three separate divine beings in this story, as explained above.