The Israeli Rabbanut sent out a clarification of how they feel people should deal with the various aspects of Yom HaAtzmaut and Yom Yerushalayim due to them being pushed up and not celebrated on their original dates.
The Rabbanut writes:
As is known, this year Yom HaAtzmaut falls out on Friday. In order to prevent desecration of Shabbos that would be caused by the various events as a result of being so close to Shabbos, it has been decided to make the celebrations and ceremonies earlier to Thursday 4 iyar.
Yom HaZikaron is therefore also made earlier, pushed up to Wednesday 3 Iyar.
With that, the minhag to not say tachanun on Yom HaAtzmaut will be followed this year as well as it is every year on the original date of Yom HaAtzmaut, on Friday the 5th of Iyar. The change in the date is only relevant to the various events, and not regarding the tefillot.
Yom Yerushalayim, which is on 28th of Iyar, this year falls out on Sunday. The Rabbanut has decided that this day also, in order to avoid potential desecration of Shabbos when preparing for the various events, events will not be held on Saturday night but will be delayed until Sunday.
However, also on this day, the prayers of Yom Yerushalayim are not changing. The tefilla said on the night of Yom yerushalayim will still be said on Saturday night, and the prayers said by day will be said Sunday morning as every year.
The Rabbanut makes no reference to Hallel or other special prayers some people say.
Because the Rabbanut statement of clarification was so unclear, the Rabbanut issued another clarification shortly after the first.
In this reclarification, the Rabbanut explains that, in their opinion, Tachanun should not be said today at Mincha of Yom HaZikaron, Thursday being Yom HaAtzmaut, and on Friday 5 Iyar.
The Rabbanut still said nothing specifically about Hallel and the other prayers.
Rav Ovadiah Yosef writes that although the Jewish people experienced a miracle with the establishment of the State of Israel, the miracle did not include all of the Jewish people and, therefore, Hallel should not be said with a bracha. Nevertheless, he writes that “If the congregation wishes to say Hallel without a blessing after the prayer service, they should not be prevented.”
Most poskim in the chareidi world have been clear in paskening that Hallel should not be said.