The fifth of Iyar can fall out on a Monday, Wednesday, Friday, or Saturday (Shabbat). When it falls out on a Friday or Shabbat, there is good reason to fear that the celebrations and ceremonies will cause public desecration of the Sabbath. Therefore, it was decided – at the request of the Chief Rabbinate – that whenever Yom HaAtzma’ut falls out on a Friday or Shabbat, the holiday is celebrated on [the previous] Thursday (the 3rd or 4th of Iyar). Eventually, the Rabbis realized that even when Yom HaAtzmaut falls out on a Monday, the preparations for Yom HaZikaron (Memorial Day), which begins on Saturday night, cause many Jews to violate the Sabbath. Therefore, it was decided – at the request of the Chief Rabbinate – that both of these special occasions be postponed by a day, establishing Yom HaZikaron on the fifth of Iyar and Yom HaAtzmaut on the sixth of the month. In practice, then, on three of the four days on which Yom HaAtzmaut can fall, we celebrate it either before or after its genuine date.
We find a similar concept elsewhere. Out of concern that one might carry a shofar or lulav on Shabbat, the Sages canceled these mitzvot. Therefore, when Rosh HaShanah falls out on Shabbat, we do not blow the shofar that day, and when the first day of Sukkot falls out on Shabbat, we do not take the Four Species. Thus, the Sages canceled biblical commandments in order to avoid Sabbath violation, but we may not change the actual date of a holiday since it is written explicitly in the Torah. Rabbinically-ordained holidays, however, may be postponed, or observed earlier. For example, when Purim falls out on Shabbat, we read the Megillah and give gifts to the poor on Friday, read the special Torah reading and say Al HaNissim on Shabbat, and eat the festive Purim meal and send portions of food to others on Sunday (Shulchan Aruch. Orach Chaim 688:6; Mishna Berura 18; below 17.5). And when Tish’a B’Av falls on Shabbat, we postpone the fast until Sunday (see Shulchan Aruch. O.C. 551:4, 554:19).
The same is true of Yom HaAtzmaut – it all depends on how the holiday was instituted. Whichever day the representatives of the people and the Chief Rabbinate decide is the day to celebrate the establishment of the State is the day that we must thank Hashem for His salvation10.
It is interesting to note that the declaration of independence took place earlier than originally planned, in order to prevent Sabbath desecration. After all, the British Mandate ended on Friday night, [May 14, 1948], at midnight, but the heads of the People’s Council did not want to declare statehood amidst Sabbath desecration, so they moved the declaration up to Friday afternoon, the fifth of Iyar.
 This has been the Chief Rabbinate’s position throughout. True, in 5741 (1981), Rabbi Goren thought that one should say Hallel on the fifth of Iyar that falls on Shabbat, reasoning that whatever does not entail Sabbath desecration should be done in its proper time, as is the case on Purim (Torat HaShabbat VeHaMo’ed). Nevertheless, all the other Rabbis held that no distinction should be made, as Rabbi Ariel explains in Ohalah Shel Torah 73. Rabbi Ariel assumes that we omit tachanun on the fifth of Iyar (if it falls on a Friday), similar to Tish’a B’Av that falls out on Shabbat: even though we postpone the fast until Sunday, the day retains its status of Tish’a B’Av with regard to certain laws (see there). If, however, the fifth of Iyar falls out on Shabbat, we say Av HaRachamim, because of the misfortunes that occurred in Iyar (this concludes Rabbi Ariel’s comments). It seems to me that we should, nevertheless, omit Tzid’katcha from the Minchah service. It also seems – and so decided the Chief Rabbinate in 5764 (2004) – that when the fifth of Iyar falls on a Monday, pushing off Yom HaAtzmaut until Tuesday, we do not say Tachanun on Monday. In his work, HaRabbanut HaRashit (pp. 898-99), Rabbi Shmuel Katz explains that even Rabbi Goren originally held that Hallel should be said on the day that was chosen for the general celebrations, as opposed to Shabbat. He changed his mind only in 5741 (1981). In footnote 33, Rabbi Katz relates, in the name of Rabbi Alfasi, that in 5761 (2001) the members of Rabbi Goren’s synagogue, Komamiyut Avraham, followed the prevalent custom not to say Hallel on Shabbat, because there were those who heard from Rabbi Goren that the general public did not accept his ruling on this matter.
Rabbi Kook writes in Mitzvat Re’iyah (Orach Chaim 688:1) that the Sages instituted two days of Purim in order to distinguish between a biblical commandment, which has one fixed time for everyone, and a rabbinic one, which is observed at different times in different locations. Perhaps this explains why the Rabbis instituted two levels of Mehadrin with regard to the Chanukah candles, something we do not find in relation to biblical commandments, whose laws are fixed. Based on this, we can say that it is fitting for Yom HaAtzmaut not to have a fixed date, seeing that it is a rabbinic enactment.