Talmud Bavli, Beitzah
Beitzah 4b
The Gemara discusses the institution of Yom Tov Sheni (the second day of Yom Tov in the Diaspora), explaining that even after the calendar was fixed, the Rabbis maintained the practice so as not to disparage the custom of the fathers ('minhag avoteihem').
וְהַשְׁתָּא דְּיָדְעִינַן בִּקְבִיעָא דְיַרְחָא, מַאי טַעְמָא עָבְדִינַן תְּרֵי יוֹמֵי? מִשּׁוּם דִּשְׁלַחוּ מִתָּם: הִזָּהֲרוּ בְּמִנְהַג אֲבוֹתֵיכֶם בִּידֵיכֶם, זִמְנִין דְּגָזְרוּ הַמַּלְכוּת גְּזֵרָה וְאָתֵי לְאִקַּלְקוּלֵי.
The Gemara asks: And now that we know the determination of the first day of the new month, what is the reason that we observe two Festival days in the Diaspora? Because they sent a warning from there, from Eretz Yisrael: Although now there is a fixed calendar and there is no uncertainty, be careful to observe the custom of your fathers that you received, because at times the monarchy will issue decrees of persecution restricting Torah study and the fixed calendar may be forgotten. And the people will come to have their proper observance of the Festivals be disrupted again. However, the fundamental halakha is that the observance of two Festival days is based on uncertainty.
Why it matters — Provides the Talmudic rationale for why Yom Tov Sheni was kept even after the calendar was standardized — essential for understanding whether an Israeli abroad is bound by it.