This is a bit of a difficult question to answer, because people (in Judaism, anyway) don't generally say "I am starting practice x for y reason". And part of the problem will be a definition one--the Holocaust was a massively transformative event that reshaped Jewish history. Really, any change to Jewish practice since 1945 can be said to be a result of the Holocaust, since it changed Jewish demographics in very important ways.
In terms of actual concrete changes, there aren't many direct ones. The only one I can think of is Yom HaShoah, which is the Israeli Holocaust remembrance day. It is a memorial day of sorts in Israel, and is mostly observed in secular ways. There are memorial ceremonies, and the most notable memorial is the sounding of air raid sirens for a national moment of silence, which also happens on the military memorial day in Israel (I suggest taking a look on youtube for what this looks like--it's very much a hair-on-back-of-neck-raising, goosebump-inducing thing to watch and hear). Also festive things are generally not done then.
But, you asked about religious practice. There are certain segments of Judaism, though, which observe it in traditional religious mourning ways. There are people who fast on that day, or say additional things in the liturgy, etc. But it hasn't gained universal acceptance. It is observed in many Conservative and Reform communities in some way, and some left-leaning Orthodox communities I believe too. Part of the problem is that minor fast days are not widely observed outside Orthodoxy (and more observant Conservative Jews), so adding another day to the roster is not going to gain widespread acceptance. It is also widely observed with ceremonies and the like, rather than fasting and changes to the liturgy.
In terms of why it hasn't gained wider acceptance in Orthodoxy, part of the problem is that the date is during the month of Nisan, which is traditionally a time of celebration. Another part of the problem is that it was instituted by Israel's secular government, and the selection of the date reflected their ideological bent, which is uncomfortable for the non-Zionist spectrum of Orthodoxy. Another problem is that it historically did not include all Jewish communities, unlike the more traditional mourning observances. Probably the biggest problem is a reluctance to add new holidays in the modern era, both festive and mourning. There is some precedent for previous generations to add holidays for important dates in their community, usually fasts for dates of major persecutions, or a "second Purim" for being saved from persecution. But people generally don't observe those today, and they don't seem to have "taken" outside the communities and eras they developed in. I think it's more common to have some sort of ceremony for it in Orthodoxy, but those are the reasons religious observances don't seem to be very popular. There's a great deal of reluctance to make (obvious) liturgical changes at all, partly as a counter-reaction to Conservative Judaism's changes to the liturgy over the last 50 years or so. Somewhat more popular is adding liturgical poems about the Holocaust to existing days of mourning, which has been done through history to an extent, and those poems generally did stick around.
Really, part of it is that Judaism has dealt with this already. There've been a number of persecutions which have changed Jewish history in profound ways, and we don't have holidays for them. The inbuilt way Judaism tends to memorialize persecution religiously is by adding extra motivation to the traditional mourning dates for the destruction of the Temple, or by adding liturgical poems to those dates.
If you want a non-random sampling, none of the dozen or so Orthodox prayer books I have include any particular liturgical observance for Yom HaShoah. Even the Koren siddur, which includes special liturgical additions for Israeli Independence Day and Israeli Solider's Memorial Day doesn't have anything for Yom HaShoah. The old Sim Shalom (Conservative prayer book) I believe includes a memorial prayer for it somewhere, and has readings about the Holocaust (which is a good way to mildly traumatize children who come to synagogue)--that's just from memory, I don't have one to reference. I believe the new one has more liturgical things for it, but I don't know offhand. IIRC it does mention there are congregations where people are fasting for it. Theoretically I could imagine people saying the Martyr's Prayer on the nearest Shabbat, but I'm not sure anybody does. There are places out there in Conservative Judaism where some sort of Holocaust readings are added to the Martyrology in Yom Kippur (a narrative about a bunch of martyred Rabbis in the Roman period). IIRC that practice itself was a reaction to persecution in Europe--as I said before, extra motivation of existing mourning/destruction things in religion is a pretty common thing historically.
In terms of non-obvious things resulting from it, it's really hard to pick. The growth of Orthodoxy in the US (especially more traditional forms than had previously been popular) is a major result, but that's not a religious practice per se, but rather the demographics of who's observing what practices. The accent most Israelis have, and some other Israeli things, are due to the influx of German Jews in the 1930s but that's not a religious practice either.
I hope this answers your question.