A good resource for this topic is Darwell Stone’s A History of the Doctrine of the Holy Eucharist.
At the last supper, after Our Lord uttered the words “this is My Body,” he continued on and said “Do this in remembrance of me” to the great delight of memorialists. Memorialists—following Huldrych Zwingli—when faced with the phrase “this is my body” point to the next phrase, “Do this in remembrance of me.” This phrase “in remembrance of me” acts as a bulwark of defense for the memorialists in their denial of real presence and is used by protestant bodies at large to deny a sacrificial eucharist.
How are we to respond to such claims made by memorialists? Are we to deny fifteen centuries of the teaching of the Fathers of the church and the liturgy which they formed? It is my conviction that the exact opposite is true, that in fact “Do this in remembrance of me” does not signify a bare commemoration of what happened 2000 years ago but is a true sacramental participation in the one sacrifice of Christ, by which life is given to us and sin is blotted out. This phrase “do this in remembrance of me” was actually—as the Fathers put it—the institution of the “bloodless sacrifice of the New Covenant." Rather than teaching the bare memorialism of Zwingli, the words of institution teach the catholic doctrine of the eucharist, as accepted by the Anglican, Roman, and Eastern churches.
In order to prove this, we will look at 4 phrases within the context of the words of institution and show how they all connect and witness to this being the institution of “the bloodless sacrifice” and not a “bare memorial.” First, διαθήκη (covenant), then, ποιεῖτε (do), third, ἀνάμνησιν (remembrance), and fourth, ἐκχυννόμενον (Being poured out).
This word is most often translated as “covenant,” or in usage more familiar to us “testament” (it is where our phrase “new testament” comes from). It is found in the phrase “this is my blood of the Covenant (διαθηκη),” and in another account “This cup is the New Covenant (or testament, διαθηκη) in my blood." This language of “covenant/διαθήκη” brings to mind the Old covenant sacrificial feasts by which the Israelites ate “Old Covenant” sacrifices and through it achieved remission of sin and blessing sacramentally. It can be naturally understood through the lens of Hebrews 9:15ff with the language of the “blood of the New Covenant”:
"And for this cause he [Christ] is the mediator of the new testament, that by means of death, for the redemption of the transgressions that were under the first testament, they which are called might receive the promise of eternal inheritance….neither the first testament was dedicated without blood. For when Moses had spoken every precept to all the people according to the law, he took the blood of calves and of goats, with water, and scarlet wool, and hyssop, and sprinkled both the book, and all the people, Saying, This is the blood of the testament which God hath enjoined unto you."
Interestingly, though the ratifying of the covenant is by blood (sprinkling it on the people, altar, and book of the covenant), this was followed with a sacrificial meal with God himself,
"Then went up Moses, and Aaron, Nadab, and Abihu, and seventy of the elders of Israel: And they saw the God of Israel: and there was under his feet as it were a paved work of a sapphire stone, and as it were the body of heaven in his clearness. And upon the nobles of the children of Israel he laid not his hand: also they saw God, and did eat and drink."
The ratification of the covenant was effected through both the application of the sacrificial blood and a meal with God himself, both of these ideas in the ratification of the covenant are brought together by bringing “the blood of the covenant” into the sacramental meal itself.
It is impossible to escape the fact that for Jews of the 1st century the idea of the “blood of the covenant” was explicitly sacrificial, even when brought into the event of a meal (though as we have seen it is a natural place for it). The idea of “blood” and “covenant” brought together means sacrifice each and every time. Therefore, we ought to catch these sacrificial overtones which were present at the last supper. Our continuing investigation will strengthen this conclusion.
Though the meaning of “do” seems pretty straightforward (i.e. “perform this action”) there is much more nuance that can be placed behind it, revealing a sacrificial context that is spoken of. Both the Greek word ποιεῖτε and its' Hebrew counterpart עָשָׂה in certain contexts have the sense of “offering a sacrifice.” A few examples of this are,
“The one lamb thou shalt offer (“Do”) in the morning; and the other lamb thou shalt offer (“Do”) at even.”
“and Moses said unto Aaron, Go unto the altar, and offer (“Do”) thy sin offering, and thy burnt offering, and make an atonement for thyself, and for the people: and offer (“Do”) the offering of the people, and make an atonement for them; as the Lord commanded.”
“I will offer (“Do”) unto thee burnt sacrifices of fatlings, with the incense of rams; I will offer (“Do”) bullocks with goats. (see more examples below)
Therefore, as we see to “offer” is well within the normal, semantic domain of the word “ποιεω.” So, if it can be shown that the context of the usage of “ποιεω” is sacrificial, then to interpret the verb with a sacrificial connotation according to its context is fitting. This sacrificial context has been shown and will continue to be shown.
This is the word, “remembrance” or “in memory” that so often causes the idea of a bare commemoration in the eucharist, with no connection to sacrifice (except that it makes us think of it). This word occurs five times in the Septuagint (the Greek version of the Old Testament). Here they are (borrowed from Stone):
“And you shall put pure frankincense on each row so that it shall be for the bread as a memorial (“αναμνεσις”) offering, an offering made by fire for Yahweh.”
“And on the day of your joy and in your appointed times, at the beginning of your months, you will blow on the trumpets in addition to your burnt offerings and in addition to the sacrifices of your fellowship offerings. And they will be as a memorial (“αναμνεσις”) for you ⌊before⌋ your God; I am Yahweh your God.”
“A psalm of David. To bring to remembrance (“αναμνεσις”)”
“A psalm of David. To bring to remembrance (“αναμνεσις”)”
“But they were troubled for a small season that they might be admonished, having a sign of salvation, to put them in remembrance (“αναμνεσις”) of the commandment of thy law”
Passages 1 and 2 clearly are referring to a sacrificial memorial that is set before God. Passage 5 refers to a mere memorial that is made before men. On passages 3 and 4, Stone comments,
“The third and fourth passages are not without share in the obscurity which surrounds the titles of the Psalms; but the probability is very strong that a memorial before God is denoted. The best commentators explain the title of these two Psalms as a liturgical note signifying that the Psalms were to be used in connection with the offering of incense, or, as appears to be more probable, the offering of the Azkara, as the portion of the meal offering mixed with oil and burnt with incense on the altar (Lev. ii. 2) and the incense placed on the shewbread and afterwards burnt (Lev. xxiv. 7) were technically called in the Levitical ritual; and these are among the many passages in which the marginal renderings of the Revised Version preserve translations more acceptable to the best Hebrew scholars than those printed in the text of that version. Moreover, on the less likely hypothesis that the titles of these Psalms refer to their contents, not to their liturgical use, the sacrificial meaning of a memorial before God would not be absent. "His broken- hearted faith," wrote Dr. Kay, explaining the title in reference to the contents of the Psalm, " is presented to the Lord like the azkaraha-frankincense of the meat-offering, burnt with fire."
What we have here with the usages in the Septuagint are four uses describing a sacrificial memorial before God, and only one which denotes some idea of a bare memorial before men with only a notional use. Lastly, we have Hebrews 10:3 (the only use of αναμνεσις outside of the words of institution), which states “In those sacrifices there is a remembrance (“αναμνεσις”) made of sins year by year.” Again, an “αναμνεσις” is connected with a sacrifice (though in this instance it is an “αναμνεσις” for men not God). After looking at the various usages of an “αναμνεσις,” it becomes most likely, due to its connection with blood, a covenant, and the possible sacrificial usage of “offer,” that in all probability, this is speaking of a sacrificial memorial which is placed before God, to be a “memorial offering.”
This word is translated as “shed” or “poured out” in the gospel narratives, connecting grammatically back to “cup’ or “blood.” This word clearly refers to a sacrificial concept, that of the pouring out of blood at the bas of the altar, or the drink offering which is poured out before the Lord (just as St. Paul describes himself in his second epistle to St. Timothy).
Side Note: The Matter of the Sacrament
As a side not we may also point to the fact that the “matter” of the sacrament (bread and wine) point to it being the bloodless sacrifice of the New Covenant. From the very beginning bread and wine were offered as sacrifices to the Lord as seen in Melchizedek. Later the same is seen in Jewish offerings of flour and wine, and the same trend is even seen in Roman and Greek sacrificial practices. To Christians of the first century, bread and wine would naturally have suggested a sacrifice to them, whether they came form Jewish or Gentile backgrounds.
In conclusion, this phrase of “in remembrance of me” does not denote a mere, bare memoralism by which we merely think of something that happened 2,000 years without any sacramental participation, or true propitiation. In fact it is quite the opposite, the language and even the matter used by our Lord is exceedingly sacrificial. In light of this, to synthesize the linguistic conclusions reached in this extremely short overview, we may rightly interpret these words of institution as “Offer this as a memorial [offering/sacrifice] of me”.”
 See: Agreed_Statement_on_Eucharistic_Doctrine  “In the Pentateuch alone there are dozens of examples. Brenton's English translation of the Septuagint translates poieo as "sacrifice" in Exo 10:25; Lev 14:19a; 22:24; 23:19; Num 15:8; Deut 12:27. It uses "offer" in Exo 29:38; Lev 9:7, 16; 14:30; 15:15, 30; 16:24; 23:12; Num 6:11a, 16, 17; 8:12; 15:3, 5, 6, 14, 24a; 28:4, 8, 15, 21, 24, 31; 29:2, 39.”  A history of the doctrine of the holy eucharist: Stone, Darwell