Is Water Baptism Necessary Before Partaking in Lord's Supper

"Last Supper" by Giovanni Domenico Tiepolo (1750)
Yes, it is. Water baptism identifies one with the redemption work of Jesus Christ, with His death, burial, and resurrection. It is anticipated of visible identification with Christ and His Church. Every person has the personal responsibility to examine him/herself before deciding to partake in the Lord's Table. The Bible makes it clear that those who chose not to be baptized were rejecting the counsel of God (Lk.7:30). In a mixed congregation, it is not possible to always know who is worthy to partake of the Table; however, the minister must encourage only those who have been baptized for remission of sins (not just as a ritual but by faith in Jesus Christ) to partake of the Table.

Before Jesus sat down to dip bread in the cup, He washed His disciples' feet. He makes the statement that they are already "washed" and only need feet to be washed. Of course, this may not explicitly/only refer to their baptism, for they were washed by the Word. However, the element of water tying both the events of water baptism and washing of feet signifying the greater spiritual truth of wholistic salvation (of body-soul-spirit) cannot be ignored. They need to wash before they sit at the Table.

Peter said to Him, “You shall never wash my feet!”
Jesus answered him, “If I do not wash you, you have no part with Me.” Simon Peter said to Him, “Lord, not my feet only, but also my hands and my head!” Jesus said to him, “He who is bathed needs only to wash his feet, but is completely clean…” (John 13:8-10)
Following are examples of the sequence:
FIRST: "...all were baptized into Moses in the cloud and in the sea,"
THEN: "...all ate the same spiritual food, and all drank the same spiritual drink. For they drank of that spiritual Rock that followed them, and that Rock was Christ." (1Cor 10:2-4)

FIRST: "Then they that gladly received his word were baptized: and the same day there were added unto them about three thousand souls."
THEN: "And they continued stedfastly in the apostles' doctrine and fellowship, and in breaking of bread, and in prayers."
Example from Church History
A glimpse into the Early Church practice of the Lord’s Supper is obtained from Justin Martyr who lived between 103-165 AD; in his First Apology, he wrote:
Chap. LXV.—Administration of the Sacraments. But we, after we have thus washed him who has been convinced and has assented to our teaching, bring him to the place where those who are called brethren are assembled, in order that we may offer hearty prayers in common for ourselves and for the baptized [illuminated] person, and for all others in every place, that we may be counted worthy, now that we have learned the truth, by our works also to be found good citizens and keepers of the commandments, so that we may be saved with an everlasting salvation. Having ended the prayers, we salute one another with a kiss. There is then brought to the president of the brethren bread and a cup of wine mixed with water; and he taking them, gives praise and glory to the Father of the universe, through the name of the Son and of the Holy Ghost, and offers thanks at considerable length for our being counted worthy to receive these things at His hands. And when he has concluded the prayers and thanksgivings, all the people present express their assent by saying Amen. This word Amen answers in the Hebrew language to γένοιτο [so be it]. And when the president has given thanks, and all the people have expressed their assent, those who are called by us deacons give to each of those present to partake of the bread and wine mixed with water over which the thanksgiving was pronounced, and to those who are absent they carry away a portion.

Chap. LXVI.—Of the Eucharist. And this food is called among us Εὐχαριστία [the Eucharist], of which no one is allowed to partake but the man who believes that the things which we teach are true, and who has been washed with the washing that is for the remission of sins, and unto regeneration, and who is so living as Christ has enjoined. For not as common bread and common drink do we receive these; but in like manner as Jesus Christ our Saviour, having been made flesh by the Word of God, had both flesh and blood for our salvation, so likewise have we been taught that the food which is blessed by the prayer of His word, and from which our blood and flesh by transmutation are nourished, is the flesh and blood of that Jesus who was made flesh. For the apostles, in the memoirs composed by them, which are called Gospels, have thus delivered unto us what was enjoined upon them; that Jesus took bread, and when He had given thanks, said, “This do ye in remembrance of Me, (Luke 22:19) this is My body;” and that, after the same manner, having taken the cup and given thanks, He said, “This is My blood;” and gave it to them alone.