In Mark 7:1ff it is written that the Pharisees had a tradition by which they invalidiated the word of God, a tradition that said it was requisite for people to religiously (not merely hygienically) wash their hands each time before sitting down to eat. The Pharisees said this practice was based on the oral teachings of Moses, but Jesus said it was vanity. Jesus cited the written words of Moses and Isaiah to show how fundamentally out of step the Pharisees were with the truth of God.
What the Pharisees did had a semblance of truth to it. The Mosaic law did prescribe certain cleansing rites as necessary to certain eating rites (e.g., Num. 18:8-20, and Lev. 21-22, both pertaining to priests; and Num. 9:1-14, and 19:11-22, pertaining to the people); this is one reason why their extra-Scriptural tradition was so easily pawned off on God's people as a pious activity. The problem is that the washings that were required by the Mosaic Law in order to participate in certain sacrificial or festal meals were washings and meals instituted by God for a religious purpose. The Pharisees had elevated their common meals to the level of those that were divinely instituted by God Himself; they were honoring God with their lips but themselves in their hearts and actions.
When it comes to Passover and the Lord's Supper, we have two eating ordinances, one in the Old Testament and one in the New Testament, both of which were instituted by God Himself, and both, as the following facts establish, are said to call for cleansing/washing. Such washing signifies the need for regeneration and the forgiveness of sin in order to approach the Lord for continual nourishment. The cleansing rite of all cleansing rites in the Old Testament was Circumcision; in the New Testament, all cleansing rites have been reduced to Christian Baptism.
1. All of the so-called branches of the Christian Church - Roman, Orthodox, and Protestant - hold baptism to be the sign of union with Christ, of cleansing by His blood and Spirit, and, hence, the sign of admission into the Church, the body of Christ. As such, baptism has historically been held to precede admission to the Lord's Table.
2. All Protestant denominations - whether Baptist, Lutheran, Methodist, Presbyterian, etc. - hold baptism to be prerequisite to a worthy reception of the Lord's Supper.
1. Christ gave the great commission and included in it the command to baptize those who became His disciples through faith (Matt. 28:18-20; Mk. 16:16). This was to be followed by "teaching them to observe whatsoever things I have commanded you", and the Lord's Supper is surely to be included among the things He commanded ["Take, eat...Take, drink..." (Matt. 26:25-28); and "This do in remembrance of Me." (1 Cor. 11:24)].
2. The apostles baptized believers (and their households) immediately upon profession of faith (Acts 8:12-13, 35-38, 9:17-19, 10:44-48, 16:14-15, 22-34, 18:8, 19:1-7, 22:12-16). Only after this do we read of ongoing commitment to apostolic teaching, fellowship, participation in the Lord's Supper, etc. This was true from the first, for on the day of Pentecost we read:
"Peter said to them, 'Repent, and each of you be baptized in the name of Jesus Christ for the forgiveness of your sins; and you will receive the gift of the Holy Spirit,....so then, those who had received his word were baptized; and that day there were added about three thousand souls. They were continually devoting themselves to the apostles teaching and to fellowship, to the breaking of bread and to prayer....Day by day continuing with one mind in the temple, and breaking bread from house to house, they were taking their meals together with gladness and sincerity of heart (Acts 2:38-46).
Note the order: 1) Faith in the Gospel message; 2) Baptism into the body of Christ; and 3) Acceptance to the fellowship meal. This order is found repeatedly in conversion account after conversion account throughout the New Testament (faith --> baptism --> fellowship (inclusive of the fellowship meal)).
3. As a lead in to his discussion of how to properly administer and receive the Lord's Supper, which is found in 1 Corinthians 11, the apostle Paul says: "I do not want you to be unaware, brethren, that our fathers were all under the cloud and all passed through the sea; and all were baptized into Moses in the cloud and in the sea; and all ate the same spiritual food; and all drank the same spiritual drink, for they were drinking from a spiritual rock which followed them; and the rock was Christ" (1. Cor. 10). Though Paul passes on to make applications of this that are relevant to the abuses prevalent in the Corinthian church, we should not miss the equally applicable observation as it applies to the question of the order in which baptism and the Lord's Supper are to take place: Baptism first (Red Sea crossing), Lord's Supper second (eating "spiritual food" and drinking "spiritual drink" in the desert).
4. The people Paul was speaking to in 1 Corinthians 11 regarding worthy reception of the elements were the same body of people spoken of in Acts 18:1-11, "...and many of the Corinthians hearing, believed and were baptized" (vs. 8). It is to baptized believers that Paul is giving instruction on how to receive the body and blood of Christ in a worthy manner.
5. In order to partake of the Supper in a worthy manner, Paul calls on each individual to "examine himself, and in so doing he is to eat of the bread and drink of the cup" (1 Cor. 11). In a later letter to the Corinthians the apostle Paul elaborates on such examination: "Test yourselves to see if you are in the faith; examine yourselves" (2 Cor. 13:5). If those who take the Supper must already be "in the faith", then they must also already be baptized, for baptism is to follow hard on the heels of faith (See point 2 above).
1. The following considerations presupposes that baptism is the New Covenant counterpart of circumcision, and that what the Old Testament says applies to us, being just as "inspired of God and profitable, for doctrine, for reproof, for correction, and for instruction in righteousness,..." as it was on the day it proceeded from the mouth of God.
In light of this, it may be confidently inferred that baptism must precede admission to the Lord's Table, for it follows by a (theo-)logical necessity from Exodus 12.
"The Lord said to Moses and Aaron, 'This is the ordinance of the Passover; no foreigner is to eat of it; but every man's slave purchased with money, after you have circumcised him, then he may eat of it. A sojourner or hired servant shall not eat of it. It is to be eaten in a single house, you are not to bring forth any of the flesh outside the house, nor are you to break any bone of it. All the congregation of Israel are to celebrate this. But if a stranger sojourns with you and celebrates the Passover to the LORD, let all his males be circumcised and then let him come near to celebrate it; and he shall be like a native in the land. But no uncircumcised person may eat of it" (Ex. 12:43-48).
We see an illustration of this in the history of God's people under Joshua (Read: Josh. 5:1-15)
2. Baptism is a picture of newness of life; the Lord's Supper is a picture of feeding on Christ. Just as life must precede feeding, so by analogy new life must precede spiritual feeding on Christ.
3. Baptism is a sign of cleansing representing to us both the blood of Christ, by which our sins are washed away, and the Spirit of Christ, by which we are cleansed of our corruption (beginning with regeneration when the Spirit gives us a new heart). According to Scripture, no unclean person may eat the Lord's Supper. Since baptism signifies this requisite cleansing, it must precede approaching the table.
 For a defense of this, see my: "Circumcision is Now Baptism".