(Brief excursus here: the entire book of Leviticus "happens" in the space of a couple of verses in Exodus 40. Otherwise, the chronology of the books, read straightforwardly, looks off. This helps explain why Nadab and Abihu meet the end they meet in Lev. 10, but that is another story for another day.)
In Lev. 9, the priesthood's consecration and orientation is completed: Aaroan and Moses prepare sacrifices and burn them on the altar. At this point, Moses and Aaron retreat: "And Moses and Aaron went into the tabernacle of meeting, and came out and blessed the people [the text of this blessing isn't revealed until Num. 6: time in the Torah is wibbly-wobbly]. Then the glory of the Lord appeared to all the people [situating us firmly in Exodus 40, chronologically] and fire came out from before the Lord and consumed the burnt offering and the fat on the altar. When all the people saw it, they shouted and fell on their faces" (Lev. 9:23-24). When the Glory-Cloud/Spirit descends, the heavenly fire comes with it, enflaming the altar with the proper, uncreated fire from God.
While I'll need to do more work to investigate, the connection between what is happening here and the Day of Pentecost seems solid. The people of God, led by the Apostles, are the new Temple and the new altar upon which "living sacrifices" (Rom. 12) are made, which is our Word-infused (logikos) act of worship.