Talmud Bavli, Beitzah
Beitzah 4b
The Gemara discusses the nature of Yom Tov Sheni shel Galuyot and the principle that even though the Sages knew the calendar, they maintained the second day as a safeguard ('minhag avoteichem b'yedechem'), treating it as a binding rabbinic enactment.
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 — The Talmud's reasoning for why Yom Tov Sheni remains binding even after the calendar was fixed is directly relevant to whether an Israeli visiting abroad is bound by it.