What is the biblical basis for classifying the Eucharist as an offering or sacrifice?

It's anachronistic to ask the question "What is the biblical basis for this belief" because the belief came before the Bible itself. The Bible was written with the conviction in mind that Jesus had sacrificed Himself on Calvary and He had instituted a sacrament (liturgical celebration) equivalent to the Jewish Passover ceremony (seder meal) commemorating that sacrifice. Furthermore, the apostles were convinced that they were ordained by Jesus in that very ceremony to continue reenacting it so as to allow His disciples to participate in that sacrifice analogous to the Passover sacrifice.

All four gospels make a point of specifying that the Last Supper was, indeed, a Passover seder. The seder is explicitly a meal in which a sacrificial lamb is consumed during a liturgical reenactment of God's salvation of the Jews from the slavery of Egypt. There is some controversy about a couple of ambiguities in the gospel of John, but properly understood, he too indicates that it was a seder meal. John 13:1-4 is awkwardly sequenced, but should be understood as saying after the disciples had reclined for the seder meal (which is at the very start of the first day of the seven day Passover festival), but before they began to eat, Jesus rose from the table and washed the feet of the disciples. (It wouldn't really make sense for Jesus to do it during the meal or after it since it is customary to wash feet of guests upon their arrival.) These verses are best translated by the Jerusalem Bible as follows:

It was before the festival of the Passover, and Jesus knew that the hour had come for him to pass from this world to the Father. He had always loved those who were his in the world, but now he showed how perfect his love was.

They were at supper, and the devil had already put it into the mind of Judas Iscariot son of Simon to betray him. Jesus knew that the Father had put everything into his hands and that he had come from God and was returning to God, and he got up from table, removed his outer garment and, taking a towel, wrapped it around his waist; he then poured water into a basin and began to wash the disciples' feet and wipe them with the towel he was wearing.

Later, beginning in John 13:21 the actual betrayal of Judas takes place. Note John's explanation of "That thou doest, do quickly" in verse 29: "For some of them thought, because Judas had the bag, that Jesus had said unto him, Buy those things that we have need of against the feast; or, that he should give something to the poor." Judas being sent out in the middle of the night to give something to the poor only makes sense on the night of Passover, the night when the seder is celebrated, because only on that night the gates of the temple were opened at midnight and the poor congregated to receive alms.

The other problematic passage is John 19:14, repeated in 31 and 42 "And it was the preparation of the passover" or simply "preparation". Properly understood, this means Friday of the week long festival of Passover, not the eve before the seder was eaten. Preparation was always the day before a Sabbath (Saturday). Note that in verse 31 it is specific that the following day is a Sabbath, and not just any Sabbath, but the Sabbath during a high festival week. The day before the start of a festival is erev in Hebrew, meaning eve, just like in English. Equivalent Greek terms would be hespera or opsia, or possibly paramoni which is not attested in the Bible and therefore may be only modern Greek.

Matthew 26:26-29, Mark 14:22-25, Luke 22:15-20 and 1 Cor 11:23-25 all attest that Jesus says “Take, eat; this is my body.” and “Drink of it, all of you, for this is my blood of the covenant, which is poured out for many for the forgiveness of sins.” He does this in a very special context, a paschal supper, already understood sacramentally by His Jewish audience. In Luke and 1 Cor he also says "do this in remembrance of me". Now, a paschal meal was a remembrance of God's bringing the Israelites out of Egypt, but it was never understood by the Jews as a mere symbolic remembrance. It was a liturgical recreation of the event that put the participants back in that original last night in Egypt, ready to be rescued from slavery. This was Jesus's last night before His death, and by His sacrifice the next day they were rescued from the slavery to sin. The parallelism was not lost on the apostles. It was very clear to them that he was not asking them to eat merely a symbol, but the actual food of His sacrifice, just as the lamb of the seder was the holy substance of a sacrifice. Furthermore they understood Him to be ordaining them to a new priesthood, one that required that they repeat this event over and over for the new Christian community.

John doesn't record the institution of the eucharist at the Last Supper since by the time he wrote his gospel it had already been set down in writing four times. Instead, he recorded sayings of Jesus in Chapter 6 that can only be properly understood in light of the eucharist as a sacrificial meal analogous to a Passover seder. Note verse 4: "And the passover, a feast of the Jews, was nigh." There follows the miracle of the multiplication of the five loaves and two fish in verses 5-14. The miracle of Jesus walking on the sea comes in verses 15-25.

There follows many sayings of Jesus alluding to the future sacrament of the eucharist. I will bold the key words and phrases:

26 Jesus answered them and said, Verily, verily, I say unto you, Ye seek me, not because ye saw the miracles, but because ye did eat of the loaves, and were filled. 27 Labour not for the meat which perisheth, but for that meat which endureth unto everlasting life, which the Son of man shall give unto you: for him hath God the Father sealed. 28 Then said they unto him, What shall we do, that we might work the works of God? 29 Jesus answered and said unto them, This is the work of God, that ye believe on him whom he hath sent. 30 They said therefore unto him, What sign shewest thou then, that we may see, and believe thee? what dost thou work? 31 Our fathers did eat manna in the desert; as it is written, He gave them bread from heaven to eat. 32 Then Jesus said unto them, Verily, verily, I say unto you, Moses gave you not that bread from heaven; but my Father giveth you the true bread from heaven. 33 For the bread of God is he which cometh down from heaven, and giveth life unto the world. 34 Then said they unto him, Lord, evermore give us this bread. 35 And Jesus said unto them, I am the bread of life: he that cometh to me shall never hunger; and he that believeth on me shall never thirst. 36 But I said unto you, That ye also have seen me, and believe not. 37 All that the Father giveth me shall come to me; and him that cometh to me I will in no wise cast out. 38 For I came down from heaven, not to do mine own will, but the will of him that sent me. 39 And this is the Father’s will which hath sent me, that of all which he hath given me I should lose nothing, but should raise it up again at the last day. 40 And this is the will of him that sent me, that every one which seeth the Son, and believeth on him, may have everlasting life: and I will raise him up at the last day. 41 The Jews then murmured at him, because he said, I am the bread which came down from heaven. 42 And they said, Is not this Jesus, the son of Joseph, whose father and mother we know? how is it then that he saith, I came down from heaven? 43 Jesus therefore answered and said unto them, Murmur not among yourselves. 44 No man can come to me, except the Father which hath sent me draw him: and I will raise him up at the last day. 45 It is written in the prophets, And they shall be all taught of God. Every man therefore that hath heard, and hath learned of the Father, cometh unto me. 46 Not that any man hath seen the Father, save he which is of God, he hath seen the Father. 47 Verily, verily, I say unto you, He that believeth on me hath everlasting life. 48 I am that bread of life. 49 Your fathers did eat manna in the wilderness, and are dead. 50 This is the bread which cometh down from heaven, that a man may eat thereof, and not die. 51 I am the living bread which came down from heaven: if any man eat of this bread, he shall live for ever: and the bread that I will give is my flesh, which I will give for the life of the world. 52 The Jews therefore strove among themselves, saying, How can this man give us his flesh to eat? 53 Then Jesus said unto them, Verily, verily, I say unto you, Except ye eat the flesh of the Son of man, and drink his blood, ye have no life in you. 54 Whoso eateth my flesh, and drinketh my blood, hath eternal life; and I will raise him up at the last day. 55 For my flesh is meat indeed, and my blood is drink indeed. 56 He that eateth my flesh, and drinketh my blood, dwelleth in me, and I in him. 57 As the living Father hath sent me, and I live by the Father: so he that eateth me, even he shall live by me. 58 This is that bread which came down from heaven: not as your fathers did eat manna, and are dead: he that eateth of this bread shall live for ever. 59 These things said he in the synagogue, as he taught in Capernaum. 60 Many therefore of his disciples, when they had heard this, said, This is an hard saying; who can hear it? 61 When Jesus knew in himself that his disciples murmured at it, he said unto them, Doth this offend you? 62 What and if ye shall see the Son of man ascend up where he was before? 63 It is the spirit that quickeneth; the flesh profiteth nothing: the words that I speak unto you, they are spirit, and they are life. 64 But there are some of you that believe not. For Jesus knew from the beginning who they were that believed not, and who should betray him. 65 And he said, Therefore said I unto you, that no man can come unto me, except it were given unto him of my Father.

66 From that time many of his disciples went back, and walked no more with him. 67 Then said Jesus unto the twelve, Will ye also go away? 68 Then Simon Peter answered him, Lord, to whom shall we go? thou hast the words of eternal life. 69 And we believe and are sure that thou art that Christ, the Son of the living God. 70 Jesus answered them, Have not I chosen you twelve, and one of you is a devil? 71 He spake of Judas Iscariot the son of Simon: for he it was that should betray him, being one of the twelve.

The 22nd Session of the Council of Trent on the doctrine of the Sacrifice of the Mass says that, since there was no perfect sacrifice "because of the weakness of the Levitical priesthood" (Heb. vii. 11, 18.), "another priest should rise, according to the order of Melchisedech" (Heb. v. 10.) to offer the perfect sacrifice, Jesus Christ.

The Council continues, saying Jesus Himself is the new Passover:

by those words, Do this in commemoration of me [Luke xxii. 19.] he commanded them and their successors in the priesthood to offer [them]; even as the Catholic Church has always understood and taught. For, having celebrated the ancient Passover, which the multitude of the children of Israel immolated in memory of their going out of Egypt, he instituted the new Passover [to wit], himself to be immolated, under visible signs, by the Church through [the ministry of] priests, in memory of his own passage from this world unto the Father, when by the effusion of his own blood he redeemed us, and delivered us from the power of darkness, and translated us into his kingdom. [Col. i. 13.]

Ultimately, the Eucharist is a sacrifice because it is a memorial: a memorial of an eternal sacrifice:

We carry out this command of the Lord ["Do this in remembrance of me"; cf. Luke 22:19, 1 Corinthians 11:24] by celebrating the memorial of his sacrifice. In so doing, we offer to the Father what he has himself given us: the gifts of his creation, bread and wine which, by the power of the Holy Spirit and by the words of Christ, have become the body and blood of Christ. Christ is thus really and mysteriously made present.

And this belief that the memorial is a sacrifice is based on the words of Jesus:

This is my body, which will be given for you; do this in memory of me. ... This cup is the new covenant in my blood, which will be shed for you.

(Luke 22:19–20; New American Bible, Revised Edition)

We interpret these words to mean that the bread and wine which we offer becomes the Body and Blood of Jesus Christ, and thus takes part in the sacrifice which he made and makes before the Father:

He entered once for all into the sanctuary, not with the blood of goats and calves but with his own blood, thus obtaining eternal redemption

(Hebrews 9:12, NABRE)

This latter verse is cited in a document of the U.S. Conference of Catholic Bishops titled "The Real Presence of Jesus Christ in the Sacrament of the Eucharist: Basic Questions and Answers". In Section 2, "Why is the Eucharist not only a meal but also a sacrifice?", the bishops explain that since "[Jesus'] actions transcend time, which is part of creation"¹, therefore

Jesus the eternal Son of God made his act of sacrifice in the presence of his Father, who lives in eternity. Jesus' one perfect sacrifice is thus eternally present before the Father, who eternally accepts it. This means that in the Eucharist, Jesus does not sacrifice himself again and again. Rather, by the power of the Holy Spirit his one eternal sacrifice is made present once again, re-presented, so that we may share in it.

Thus, the Biblical basis for this belief is largely present in Luke 22:19–20 and synoptic verses, as well as in Hebrews 9:11–12.

¹Support for this interpretation may be found, e.g., in Hebrews 9:11:

Christ came as high priest of the good things that have come to be, passing through the greater and more perfect tabernacle not made by hands, that is, not belonging to this creation.

(emphasis added)

The section Scriptural proof in Sacrifice of the Mass | New Advent begins by noting that the Divine institution of the Mass can be established by both the Old and the New Testament. In 2. New Testament, the section says

The main testimony of the New Testament lies in the account of the institution of the Eucharist, and most clearly in the words of consecration spoken over the chalice. (my emphasis) For this reason we shall consider these words first, since thereby, owing to the analogy between the two formulas clearer light will be thrown on the meaning of the words of consecration spoken over the chalice. For this reason we shall consider these words first, since thereby, owing to the analogy between the two formulae, clearer light will be thrown on the meaning of the words of consecration pronounced over the bread. For the sake of clearness and easy comparison we subjoin the four passages in Greek and English:

• Matthew 26:28: Touto gar estin to aima mou to tes [kaines] diathekes to peri pollon ekchynnomenon eis aphesin amartion. For this is my blood of the new testament, which shall be shed for many unto remission of sins.
• Mark 14:24: Touto estin to aima mou tes kaines diathekes to yper pollon ekchynnomenon. This is my blood of the new testament which shall be shed for many.
• Luke 22:20: Touto to poterion he kaine diatheke en to aimati mou, to yper ymon ekchynnomenon. This is the chalice, the new testament in my blood, which shall be shed for you.
• 1 Corinthians 11:25: Touto to poterion he kaine diatheke estin en to emo aimati. This chalice is the new testament in my blood.

The Divine institution of the sacrifice of the altar is proved by showing (my emphasis)

• that the "shedding of blood" spoken of in the text took place there and then and not for the first time on the cross;
• that it was a true and real sacrifice;
• that it was considered a permanent institution in the Church.

[...]

Cf. The sacrificial memorial of Christ and of his Body, the Church CCC 1362-1372.