Bible-loving Protestants often bring up the idea that nowhere in the New Testament does it speak of the Lord's Supper being a sacrifice. Orthodox Christians not only point to the long Patristic tradition that the Lord's Supper is indeed a sacrifice, but also point out a few biblical passages that clearly connect the Lord's Supper with a sacrifice.
In Hebrews 13:10 the Apostle Paul writes: "We have an altar whereof they have no right to eat who serve the tabernacle." In this passage the Apostle is comparing the altar of the Jews and the altar of the Christians and pointing out the resemblance and the difference between the two altars and two sacrifices. However, those who aren't familiar with the original Greek may assume that "altar" and "sacrifice" are independent of each other. Yet, the Greek word translated in English Bibles as "altar" implies both together. The word "θυσιαστηριον" is not simply an altar, but specifically a "sacrificial altar".
1 Corinthians 10:15-21 makes this connection even clearer. Here the Apostle Paul writes the following:
I speak as to wise men; judge for yourselves what I say. The cup of blessing which we bless, is it not the communion of the blood of Christ? The bread which we break, is it not the communion of the body of Christ? For we, though many, are one bread and one body; for we all partake of that one bread.
Observe Israel after the flesh: Are not those who eat of the sacrifices partakers of the sacrificial altar? What am I saying then? That an idol is anything, or what is offered to idols is anything? Rather, that the things which the Gentiles sacrifice they sacrifice to demons and not to God, and I do not want you to have fellowship with demons. You cannot drink the cup of the Lord and the cup of demons; you cannot partake of the Lord’s table and of the table of demons.
Here the Apostle Paul clearly connects the altar with the sacrifice of the altar, and compares the sacrifices of Israel as well as the sacrifices of the Gentiles with the sacrifice of the Christians in the Lord's Supper. He forbids those who partake of the Lord's sacrifice to partake of the sacrifice of the Gentiles when he speaks of "the Lord’s table" and "the table of demons". This table is the "sacrificial altar".