"''Concerning the appointed feasts of the LORD, which you shall proclaim to be holy convocations . . .''" (Leviticus 23:1 - 2, Holy Bible in Its Original Order - A Faithful Version (HBFV) throughout unless stated)
Notice the clear statement that Leviticus 23:2 presents. A convocation is a commanded assembly. A "HOLY convocation" is a commanded assembly based on the authority of God, who is holy. This chapter lists all of them for the entire year. When reading the chapter you will find the complete ABSENCE of any command or instruction about observing new moons.
The purpose of verse 2 in Leviticus 23 is to be the preamble to all the days in the year that God expects us to observe in this present age. Notice verse three of the same chapter.
"Six days shall work be done, but the seventh day is the Sabbath of rest, a holy convocation. You shall not do any work. It is a Sabbath to the LORD in all your dwellings" (Leviticus 23:3)
Clearly, the weekly Sabbath is one of God's holy convocations, and it is a commanded assembly. In verse 4 the statement from verse 2 is repeated, in order to also apply it to the annual Sabbath days (Leviticus 23:4). The rest of this chapter discusses the Passover, the three annual Feasts, and the seven annual Holy Days.
Leviticus chapter 23 makes very clear that ALL the days which God wants us to observe. They are the weekly Saturday Sabbath day (verse 3), the Passover (verse 5), the seven days of Unleavened Bread (the first and last days of which are a special convocation - verses 6-8), and the Feasts of Pentecost (verse 21), Trumpets (verses 24-25), Atonement (verses 27-32), Tabernacles and Last Great Day (verses 34-36).
Nowhere in the Bible, however, is there a command from God that we in this present age are to observe new moons! They simply do not feature amongst the commanded assemblies of God. Leviticus Chapter 23 is the clearest and strongest proof for the days that God has ordained for us to observe in this present age! There are no others!
In the Bible
There were specific requirements of the Old Testament temple priests on New Moon days. Trumpets were to be blown, and large numbers of sacrifices were required (Numbers 10:10, 28:11). The killing of sacrifices created a lot of meat to be eaten, so the new moon was a feast of sorts (see 1Samuel 20). Beyond the work of the priests, though, there were no statutory requirements of the people.
Special gatherings were sometimes held on the New Moons. King Saul had a special meal on the day and expected certain guests to attend. King David and Saul's son Jonathan attended these meals. Over time, Saul came to hate David and sought to kill him. David, knowing the danger of celebrating this day with Saul, decided not to attend.
"And David said to Jonathan, 'Behold, tomorrow is the NEW MOON . . . But let me go so that I may hide myself in the field until the third day at evening . . . '
"And David hid himself in the field. And when the NEW MOON had come, the king sat down to eat food" (1Samuel 20:18, 24).
Although Saul had special meals or banquets on the days of the new moons, this does not show or say it was required of Israel as a whole. God never commanded kings have special meals on these days. It seems Saul simply had a tradition of having a good meal on these days.
Those who believe in keeping these days sometimes quote Isaiah 66:23 which mentions the new moons during the millennium. Although such times are kept during the reign of Jesus on earth, there are no clear instructions from God for this present age. This approach is nothing more than reasoning that tries to INFER instructions for us today from references about the millennium. However, such inferences are not justified.
To keep track of the new moons in Biblical times was a way of keeping an accurate track of the passage of time. It was absolutely essential for keeping track of dates IF one wanted to know when the Day of Trumpets, as well as all the other Holy Days, would occur. If people, during the millennium, observe these days, the conditions will be different. Every month will always start with a new moon.
The real world
Let us look at observing the New Moons today from a practical view. When would we observe these days? Would they be based on when they occurred over Jerusalem? Would they be based on when they occur locally at whatever location on Earth where we may happen to be? The weekly Sabbath is kept based on the time at which LOCAL sunset occurs. IF we were to observe new moons based on when they occur in our locality, then we would end up keeping the Day of Trumpets (and Feast of Tabernacles) on different days from other people in different areas of the world.
In the final analysis, since there exists no direct command or instruction by God concerning the keeping of the Biblical New Moons in this age, believers are not required to observe them.