How does the burnt offering point to Jesus in the New Testament?

The New Testament reaches for the burnt offering's own vocabulary — in the original Greek — when it describes Christ's self-giving and the sufficiency of his death.

Start with the aroma-refrain that closes every burnt offering in Leviticus 1: "a burnt offering, a fire offering, a pleasing aroma to the LORD" (Leviticus 1:9, 1:13, 1:17). When Jewish translators turned the Torah into Greek centuries before Christ, they rendered that phrase with a fixed pair of their own — ὀσμὴ εὐωδίας, "an aroma of fragrance" — using it every single time the refrain closes a grade of offering.

That is the exact phrase Paul picks up when he writes about Christ:

Christ loved us and gave himself up for us, an offering and sacrifice to God, as a fragrant offering. — Ephesians 5:2

The Greek behind "a fragrant offering" is εἰς ὀσμὴν εὐωδίας — the same pair the Septuagint uses to close the Leviticus refrain. It's not a word-for-word quotation of a specific verse (Paul adds a preposition and shifts the grammar, and the phrase recurs throughout the Greek sacrificial law generally), but it is the fixed sacrificial formula itself, lifted and applied to Christ's self-giving. Paul uses the same phrase again for something much more ordinary — the Philippians' financial gift to him is called "a fragrant offering, a sacrifice acceptable and pleasing to God" (Philippians 4:18) — and even the word he chooses for "acceptable" there echoes the acceptance-language of the Greek Old Testament's Leviticus 1:3–4. A looser echo of the same word-family also appears in 2 Corinthians 2:14–16, where Paul writes of believers as "the aroma of Christ," though there the two words are split apart and varied rather than reproduced as the fixed pair.

The strongest and most direct New Testament link, though, names the burnt offering itself in order to set it aside. Hebrews 10 quotes Psalm 40, where the psalmist says God did not desire sacrifice and offering — and the Hebrew of that psalm actually contains the word olah. Hebrews renders it with the Greek word for whole burnt offerings and builds its whole argument on it:

In burnt offerings and sin offerings you have taken no pleasure... a body you have prepared for me. — Hebrews 10:5–6

The burnt offering, repeated again and again on the altar, could never finally accomplish what Leviticus 1:4 promised — full acceptance and atonement. Hebrews reads that ongoing insufficiency as the very reason the "once for all" offering of Christ's body was needed (Hebrews 10:10). Even in the Gospels, a scribe tells Jesus that loving God and neighbor matters more than "all the whole burnt offerings and sacrifices," and Jesus answers that he is "not far from the kingdom of God" (Mark 12:33–34) — the offering affirmed and relativized in the same breath.

The full study on the burnt offering traces this whole thread — from Noah's altar, through the Hebrew and Greek text of Leviticus 1, to the cross.