It seems that the more religious one is, the less likely one is to commemorate(?) the Holocaust on Yom HaShoah.
To what extent is this true? Which groups do and do not observe Yom HaShoah as their primary (or other) Holocaust remembrance day? How, and to what extent, do various groups for whom Yom HaShoah is not the primary day of remembrance acknowledge (or disacknowledge) Yom HaShoah?
Presenting a strictly Chareidi point of view on this site is like walking into a minefield, but here goes.
There were definitely religious leaders who were against instituting a special day to commemorate the Holocaust, but not all gave their reasoning.
One reason that was given came from Rabbi Gedalia Schor as quoted in Meged Givos Olam. The author there said other gedolim shared this idea as well.
When Hashem does something wonderful for us it is a kiddush Hashem, a sanctification of His name. Every time we mention it and commemorate it, there is another kiddush Hashem. The opposite is true as well. When Hashem does (or allows) something bad to happen to us it is a chillul Hashem, a desecration of His great name. Every mention of it and commemoration of it is another chillul Hashem.
A time such as Tisha Bi'Av or another period of public mourning, which was already instituted by earlier authorities, was viewed as an opening to tack this tragedy on to. This was the same approach taken by the Rishonim and Achronim concerning the tragedies in their communities. For instance the destruction to them resulting from the crusades and the Khmelnytsky Uprising, (see the Taz in O'ch 493 #2 about this). From a statistical point of view, the crusaders wiped out the same percentage of Ashkenaz Jews as the Holocaust did and no new day of mourning was instituted, but rather liturgy was added in to the Tisha Bi'Av service.
[Edit: This was the opinion of Rabbi J.B. Soleveitchik. He had a tradition from his uncle Reb Velvel pointing to wording in the liturgy of Tisha Bi'Av that that day was chosen as the singular day for commemorating national tragedies.]
It's also worth noting that the religious communities commemorate the victims almost every single week! On Shabbos when we say Av HaRachamim, they were the holy communities who's lives were taken, along with so many others. And again on all three festivals and Yom Kippur anyone in shul saying Yizkor will have said a special prayer for all victims of such acts. Any group of Jews who did not go to shul and pray will obviously have felt a commemoration sorely lacking for such a fresh wound, and will have gone looking for their own path.
So it is not the Chareidim who haven't commemorated the victims of the Holocaust, they have many such commemorations. But they have followed an apparent trend of commemorations for the last thousand years which have not singled out specific events. The one day a year where tradition does seem to single out singular events is Tisha Bi'Av, and commemorations were added.
Wikipedia deals with the question:
The Chief Rabbinate of Israel, in 1949, under the guidance of Ben-Zion Meir Hai Uziel and Yitzhak HaLevi Herzog, decided that the Tenth of Tevet should be the national remembrance days for victims of the Holocaust. The Tenth of Tevet fast commemorates the siege of Jerusalem by Nebuchadnezzar II. For this day, it recommended traditional Jewish ways of remembering the dead, such as the study of the traditional Mishnah section about ritual baths, saying Psalms, lighting a yahrzeit candle and saying Kaddish for those Holocaust victims whose date of death remains unknown. On other occasions, the Chief Rabbinate also referred to Tisha b'Av as being a date of remembrance for Holocaust victims.ref:13
The Knesset decision taken on April 21, 1951 to designate the 27th of Nisan as Yom HaShoah ignored the Rabbinate's decision from two years earlier, and the Chief Rabbinate, in turn, decided to ignore the Knesset's chosen date, one reason being the fact that Jewish law forbids fasting and certain laws of mourning during the month of Nisan, which is considered to be a month of happiness. Another view, held by influential Haredi Rabbi Avraham Yeshayeh Karelitz (known as the 'Chazon Ish'), held that Rabbinical Authorities of the modern era lacked the power to institute new days of mourning or commemoration for future generations.
While there are nevertheless Orthodox Jews who commemorate the Holocaust on Yom HaShoah, others in the Orthodox community – especially Haredim, including Hasidim – remember the victims of the Holocaust on traditional days of mourning which were already in place before the Holocaust, such as Tisha B'Av in the summer, and the Tenth of Tevet, in the winter. Several well-known Haredi rabbis, including Rabbi Michael Dov Ber Weissmandl, Rabbi Shlomo Halberstam of Bobov, Rabbi Shimon Schwab, and several others, wrote kinnot about the Shoah, to be said on Tisha b'Av.
While most Modern Orthodox Religious Zionist Jews do stand still for two minutes during the siren, in Haredi areas, no attention is given to Yom HaShoah. Most stores do not close, schools continue and most people do not stop walking when the siren sounds. The non-participation of Haredim in Yom HaShoah is one of the points which regularly causes friction between Haredim and non-Haredim in Israel, as non-Haredim consider the Haredi position of ignoring the siren and Yom HaShoah altogether to be disrespectful.
Thus, a situation has come into existence where religious forms of commemoration take place primarily on the Tenth of Tevet and on Tisha b'Av, while secular forms of commemoration take place primarily on Yom HaShoah, and either part of the population ignores the other's day of commemoration.
