One question one might ask would be "how could observing the Biblical feasts be a sin?" The answer to that is rather simple: If God gave specific instructions in his law concerning their observance, and we observe them outside of those instructions, then the observance itself is a sin. Which would also mean that if it is currently impossible to meet those conditions, then observing the feast would always be a sin. But surely this can't be the case, right?
Well, someone cited Deuteronomy 12 as the proof that this is exactly the case for all of the feasts in Torah. And since we're supposed to test everything [1 Thessalonians 5:21], this naturally includes seemingly counter-intuitive claims like the one someone made somewhere around the time of the Feast of Tabernacles. Oh, and also our own stances on the observance of these days. And whenever we find that something we are doing is a sin, we should stop doing it. The relevant passage starts in verse 5 of Deuteronomy 12:
 But you shall seek the place where the Lord your God chooses, out of all your tribes, to put His name for His dwelling place; and there you shall go.  There you shall take your burnt offerings, your sacrifices, your tithes, the heave offerings of your hand, your vowed offerings, your freewill offerings, and the firstborn of your herds and flocks.  And there you shall eat before the Lord your God, and you shall rejoice in all to which you have put your hand, you and your households, in which the Lord your God has blessed you. [Deuteronomy 12:5-7; NKJV]
It's pretty clear from these verses that everyone shall bring their offerings up to a single place where Yahweh chooses to put his name. From there to verse 28, we are warned not to make any tithes, offerings, or sacrifices in any place except that which Yahweh chooses to put his name. But that we can freely eat meat anywhere we desire so long as we do not eat the blood. Yahweh's appointed times certainly have requirements to bring sacrifices and offerings. So at the very least, if one cannot make it to whatever place Yahweh chose to put his name, then it is sinful to offer said offerings. Note that from Deuteronomy 12 alone, it would be the offerings that are sinful to offer, not the observance of the feasts. To see if there are any commandments that would make observing the feasts themselves sinful, we must look to the commandments concerning those feasts.
Before we look into these commandments, one might be tempted to simply invoke Deuteronomy 12:32 and/or Deuteronomy 5:32 and be done with it. But invoking these verses in this instance wouldn't do much good on their own. For example, it may be viewed that the claim that we should only observe the feasts in Jerusalem (the place Yahweh chose to put his name) is adding to God's commandments. Invoking Deuteronomy 12:32 in this instance would actually backfire for the person making such a claim as it would appear to others like he/she is themselves adding to Yahweh's law. Likewise, invoking Deuteronomy 5:32 and leaving it at that would simply beg the question: "How are we deviating from God's commandments?"
Anyone who invokes Deuteronomy 12:32 challenges their opponent to prove that their claims of what is or is not sinful is not an addition to or subtraction from Torah. But anyone who invokes Deuteronomy 5:32 must be able to back up their claims with Torah or will themselves be on the receiving end of the Deuteronomy 12:32 challenge. By the way, Deuteronomy 5:32 would definitely apply to observing Yahweh's feasts outside of the rules that he established in his Law. So let's now take a look at what the Law says concerning these feasts.
The first relevant passage is in Leviticus 23, which opens by declaring that the feasts contained within that chapter are Yahweh's feasts. In other words, those feasts and only those feasts are to be used to honor God. And yes, that does mean even Hanukah and Purim should not be used to honor God. In this chapter, there is no hint that there is only to be one place where the feasts are celebrated. But that doesn't mean there aren't any. It's just not here. There are however, in Exodus 12:43-49, instructions concerning who may or may not eat the Passover. But also no instructions on any one specific place.
Then we have Numbers 9:1-14, which concern how the Passover is to be kept if one is not able to do so at the appointed time. This also does not specify a single place where the Passover is to be observed. It does however, not give any requirements for keeping the Feast of Unleavened bread that comes immediately afterwards. Numbers 28-29 have the most comprehensive set of instructions concerning Yahweh's appointed times, but again, there is no indication that they are to be celebrated only in one place. But we haven't finished going through Torah to see if there are any instructions that would prohibit us from observing these times outside of that one place. However, the commands in Deuteronomy 12 concerning the offerings would definitely make impossible to fully-observe the feasts outside of that one place.
Deuteronomy 16 is where we start seeing instructions explicitly restricting where we may observe Yahweh's appointed times. So let's just quickly list-off the feast, the verses, and the type of restriction of location (if any).
- Passover [Deuteronomy 16:1-8]: Passover must be sacrificed at the location God chooses.
- Feast of First Fruits: No location restrictions.
- New Moon Feasts: No location restrictions.
- Feast of Weeks [Deuteronomy 16:9-12]: Must be observed at the location God chooses.
- Feast of Trumpets: No location restrictions.
- Day of Atonement: No location restrictions.
- Feast of Tabernacles [Deuteronomy 16:13-17]: Must be observed at the location God chooses.
Well, there we have it! Three of the feasts are only to be observed at the location that Yahweh chooses. The Passover, the Feast of Weeks (aka: Pentecost), and the Feast of Tabernacles. There is no indication that any of the special Sabbaths may be ignored if one cannot come to the location God chooses. As for the rest of the observance, it is indeed a sin to observe those three feasts outside of the place God chooses. And the place that Yahweh chose to put his name is Jerusalem. Today, most people simply cannot go to Jerusalem three times a year. But there is coming a time when all nations will go up for the Feast of Tabernacles [Zechariah 14:16-19].
By the way, I was not expecting these results in the slightest.