Will the Sacrament be administered or needed during the millennium?
During the Last Supper, Jesus foretold another time when He and His disciples would sit down together and drink the sacramental wine. “I will not drink henceforth of this fruit of the vine, until that day when I drink it new with you in my Father’s kingdom” (Matthew 26:29). The Lord revealed more to Joseph Smith about a singular sacrament meeting, when “I [Jesus] will drink of the fruit of the vine with you on the earth, and … with Peter, and James, and John, … by whom I have ordained you and confirmed you to be apostles” as well as many other key-holders (D&C 27:5,12). So Jesus will personally partake of the sacrament, but (as you asked), will the ordinance continue into the Millennium?
Paul shared a doctrine that left it ambiguous. He recounted the Last Supper, recording that he had obtained his information “of the Lord”. The purpose of the ordinance was “in remembrance of [Jesus]. For as often as ye eat this bread, and drink this cup, ye do shew the Lord’s death till he come.” (1 Corinthians 11:23-26). This suggests that the sacrament will cease once Christ returns. However, it could also be that Paul is sharing his version of the Last Supper which, when synoptized with the others, shows his is incomplete and the emphasis is on the doctrine that Christ will return and sup with the saints. We see something similar just comparing Luke with Matthew and Mark. Luke (similar to Paul) has Jesus abstaining from “the fruit of the vine, until the kingdom of God shall come” (Luke 22:18) while both Mark and Matthew show that Jesus’ point is that His wine fast will end when He returns, not that the sacrament is temporary.
Depending on your reading of Paul’s account, we may or may not always have the sacrament regularly as we do now (of course the principles involved, such as covenants, sacred remembrance, discipleship, and so forth will still be in full effect). Fortunately we have a precedent in The Book of Mormon to help us prepare for those blessed years after Jesus’ return. You may recall that Nephi taught his people that they needed to be baptized and receive the Gift of the Holy Ghost to more fully follow their Savior. This was the commandment and they had to obey for the sake of their salvation. “Behold,” writes Nephi, “this is the doctrine of Christ, and there will be no more doctrine given until after he shall manifest himself unto you in the flesh.” This sounds remarkably like Paul’s statement on the sacrament. Nephi makes it sound like baptisms and bestowal of the Holy Ghost may cease when Christ comes. But then he adds that “when he shall manifest himself unto you in the flesh, the things which he shall say unto you shall ye observe to do.” (2 Nephi 32:6). If we apply this to the sacrament, we find wisdom’s path. We will continue taking the sacrament until Christ comes and, when He does, we will listen with rapt attention to the instructions He gives us. If, as with animal sacrifice, He requires a purer offering, we will gladly give it. Or if, as with baptism in the New World, He authorizes anew and asks us to partake, then “again we[‘ll] meet around the board of Jesus, our redeeming Lord, with faith in His atoning blood.” (Hymn 186).