Short answer- if they refused to do something a guard wanted, they (almost definitely) got shot. Which is pretty much what you can assume happened in any case in which a Jewish concentration camp inmate didn't do what was asked of them by a Nazi guard, and sometimes even if they DID. They could be shot with impunity.

But shaving was generally something that was often done to prisoners in general, not just beards and not just for humiliation purposes, once they arrived and were processed at concentration camps. You may be thinking of in the time period BEFORE concentration camps, in which, indeed, shaving a rabbi's (or any bearded Jewish man's) beard was seen as a way to humiliate both him personally and the Jewish people in general. (That said, not all camps required inmates to shave- transit and labor camps were less likely to require it, and several memoirs I've seen include descriptions of bearded men.)

Humiliation in general was an extremely common tactic by Nazis once they had arrived in a place and gained control over the local Jewish population, and while any Jew could wind up the target of this degrading treatment, often communal leaders, rabbis, upper-class Jews, and identifiable Jews in general were singled out publicly. Beard-shaving was only one of many such methods. For example, following the Anschluss, many of these sorts of figures were ordered to clean the streets of Vienna (a city with a 1/6 Jewish population) with their bare hands, with Rabbi Dr Taglicht, rabbi of one of the largest synagogues in Vienna, ordered to do so while wearing his tallis (prayer shawl); other Jews were ordered to clean bathrooms while wearing tefillin (phylacteries), something which is not only degrading but a violation of Jewish law. Men and women, particularly the elderly, were forced to do mindless yet harsh physical activity for SS guards' amusement, often leading to fatal heart attacks. Suicide rates among Jews skyrocketed. Many of these humiliated Jews were fortunate enough to be able to take advantage of opportunities to leave Austria to other countries, knowing from this humiliation that there was no future for them there. Many, many others weren't so lucky.

Once the Nazis invaded Poland and WWII began, the opportunities to leave dwindled to none but the humiliations continued. As ghettos were formed, humiliation of communal leaders and rabbis remained part of the Nazi stock in trade. There is at least one well known picture of a Nazi soldier forcibly removing the beard of an elderly Jew, and memoirs include many more descriptions of such events- and even worse, descriptions of elderly men who had their beards ripped off their faces, leaving them bloody. While removing beards was often seen as a way of humiliating these men, in other cases leaving rabbis' beards as they forced them to do degrading things was its own method of humiliation- one memoir I looked at while researching this question, for example, described two rabbis with matted beards and sidelocks being forced to entertain the Nazis (and, purportedly, the Jewish laborers at the Janowska labor camp) by singing and dancing.

A responsum by Rabbi Ephraim Oshry gives a lot of insight into a typical situation and the implications for Jewish law- he was a young rabbi in the Kovno (Lithuania) Ghetto, which was a largely Orthodox ghetto in a town that had been the site of a famous yeshiva, who answered questions of Jewish law which were often heartbreaking (and about whom I wrote here). His questions and answers were buried in the ghetto and he retrieved and published them after the war. I'll quote this responsum in full here:

Question: In our suffering under the hands of the accursed Germans who daily sought to cause us physical and mental anguish by their decrees of all kinds, one of the charges they laid against us was that Jews are filthy and carry all kinds of infectious diseases. They were especially furious with any Jew who wore a beard- which most Orthodox Jews did. When the accursed evildoers saw a bearded Jew, they immediately subjected him to ridicule and harassment, with death as a possible result. Intimidated, all the bearded ghetto dwellers removed their beards- even the rabbis and Torah sages; they realized it was absolutely impossible to leave their beards intact no matter how much they suffered internally at having to give up their emulation of the "divine image."

The Jewish leaders were compelled to remove their beards for another reason: A special goal of the Germans was the destruction of the Jewish leadership. A beard was seen by the Germans as identifying a rabbi, and the Rabbiner were singled out to be hounded mercilessly and killed outright. The rabbis were consequently constrained to remove their beards in order to protect their lives.

Only two people in the ghetto retained their beards. One was the rabbi of Kovno, Rabbi Avraham DovBer Kahana-Shapira, who did not remove his beard because he was known to the Germans and stood to gain nothing by removing his beard. He therefore guarded the honor of Jewry by leaving his beard intact.

The second individual was one of the most important householders in Kovno, a chasid of Chabad, Rav Feivel Zussman, who took the risk involved and did not remove his beard. He managed to retain his Jewish pride and glory for a number of years- until the Kinderakxion of 3 and 4 Nissan 5704- March 27 and 28, 1944. On that day the Germans searched every single attic and basement, cave and tunnel, in order to find the unfortunate children whom they dragged out to be annihilated. G-d! Avenge their sacred, pure blood!

To facilitate transportation of ghetto laborers to the airfield outside Kovno, the Germans set up a camp for slave laborers outside Aleksot, a suburb of Kovno nearest the airfield. One of the Jews confined to this camp asked me what to do about his beard since in the camp there was no implement for removing his beard other than a razor blade. His question was whether he might be permitted to remove his beard with the razor because of the danger to his life.

Response: I ruled that he might shave with the razor since there is no disagreement among the authorities that it is permissible to remove one's beard with a razor to save one's life. There was not even a non-Jew available in the prisoner-of-war camp to shave this man so as to circumvent his being shaved by a Jew, whether himself or another. But even if there had been a non-Jew in the camp, it would have made no difference because the accursed Germans forbade non-Jews to help Jews in any way.

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