Leviticus 23 sets out eight appointed times and calls them the LORD's, not Israel's. Jesus kept them. The first believers were Jews who went on keeping them. Within a generation the church was mostly , and the question of what those Gentiles were obliged to do had to be answered on the spot, which is what the council in Acts 15 is convened to do.
The New Testament pulls in more than one direction, and honest readers on every side admit it. Paul tells the Colossians to let nobody pass judgement on them regarding a festival or a new moon or a sabbath, calling these a shadow of what was to come. He also keeps feasts himself, hurries to be in Jerusalem for one, and dates his travel by them. Zechariah has the nations going up to keep after the end. None of that resolves into a single rule, which is why the question is still live.
Where everyone agrees
Everyone here agrees that the appointed times were genuinely commanded, that they were given to Israel, that Christ is their substance rather than their competitor, and that no Christian is saved by keeping them or condemned for not keeping them. The disagreement is about what follows from that for a believer's practice, not about whether the feasts matter.
Fulfilled, and therefore free
The feasts pointed to Christ, he has come, and the shadow has served its purpose.
What it claims
Colossians 2:16-17 addresses this question directly and calls festivals, new moons, and sabbaths a shadow of the things to come, the substance belonging to Christ. A shadow is not a bad thing; it is simply what precedes the body casting it. Acts 15 declines to lay the law on Gentile believers, and Galatians treats a return to observing days and months and seasons as a step backward rather than forward. The church accordingly developed its own calendar, built around the and the , and treats the Mosaic feasts as fulfilled rather than binding.
Who holds it
The majority position across , , and most Protestant traditions, and the assumption behind the Christian calendar as it actually developed.
The texts it stands on
Colossians 2:16-17 A shadow of the things to come, the substance is Christ
Galatians 4:9-11 Paul's alarm at observing days and months and seasons
Acts 15:19-29 The council declines to burden Gentile believers with the law
Strongest case
It has the clearest single text on the subject, it matches what the church actually did from very early on, and it protects the freedom the New Testament fights hardest for.
Hardest question
It has to explain why Paul himself kept feasts after writing this, why Zechariah 14 has the nations keeping Tabernacles in the age to come, and why a shadow being fulfilled should mean it becomes useless rather than more legible. It also has to account for how much of the New Testament assumes readers know this calendar.
Holding the map
Notice what nobody argues. No one here says the feasts were a mistake, and no one says keeping them earns anything. The argument is about whether a fulfilled command becomes optional, becomes forbidden, or becomes clearer.
Two cautions, one for each direction, because this question has a characteristic failure on both sides. Those who keep the feasts should be wary of the drift from gift into measurement, where observance quietly becomes the mark of who is serious about God. Those who do not should be wary of treating a command God gave as merely obsolete, since none of the positions above actually says that, and dismissing the law as such is a different move from holding it fulfilled.
One practical note, since this question is usually asked by somebody about to do something rather than in the abstract. If you keep these days, keep them in a way you could explain to a Jewish neighbour without embarrassment. Say whose calendar it is. Say what you have added and why. The appointed times belong to God, but the liturgy being borrowed belongs to a living people who never stopped keeping it.
EKLI serves Christians across traditions. On questions where churches differ we explain how each one sees it and leave the deciding to you.