In the early days of Christianity, almost all believers were Jews and for a while it seems that they continued to keep the Jewish Sabbath and Old Testament festivals such as Passover and the other feasts (e.g., Acts 21:21-24)
However, once Gentiles became part of the Christian church it was necessary to decide whether keeping the various Old Testament festivals was necessary. This was addressed in a conference in Jerusalem described in Acts 15 where it was determined that none of the holy days needed to be kept (Acts 15:28-29).
Despite this decision, the keeping of holy days continued to be debated in the early church. Paul wrote to the Galatians:
But now that you have come to know God, or rather to be known by God, how can you turn back again to the weak and worthless elementary principles of the world, whose slaves you want to be once more? You observe days and months and seasons and years! I am afraid I may have labored over you in vain. (Gal 4:9-11)
Later he wrote to the Colossians
Therefore let no one pass judgment on you in questions of food and drink, or with regard to a festival or a new moon or a Sabbath. (Col 2:16)
So there is no need for the holy days specified under the law of Moses to be kept today. Nor was there a need in New Testament times, although Jews who had grown up with them continued to keep them for a time.