There is a famous Hasidic rabbi who has no facial hair, and I seem to recall that there is a famous story as to why. But I can remember neither which rabbi (or of which sect), nor the story. I do not even know if he is/was "the" Rebbe of his sect. Can someone please post information?
I cannot provide any further information, other than that I know there are color photos of him (ie., this was not centuries ago).
The Kaliver Rebbi. It was pulled out by the Nazis Yemach Shemom.
http://he.wikipedia.org/wiki/%D7%A7%D7%95%D7%91%D7%A5:Kaliver_rebbe.jpg
As others have answered, the photo of the Rebbe you are thinking of is all but certainly the Kaliver Rebbe shlita, HaRav Menachem Mendel Traub. As mentioned, the Kaliver Rebbe shlita is a Holocaust survivor, and it is my understanding (as well) that it was during this time he lost his beard. The version of the story I heard is different than Gershon Gold's or Shmuel Brill's versions of the story.
In so far as that makes three slightly, but significantly, different versions of what happened and the details are absent in the Wikipeda page about him, from the general information on the web, and perhaps most significantly from the (now defunct) website apparently established by his Chassidim, it seems to me that the details have been kept quiet out of privacy.
Perhaps until we can find more specific, better sourced, details we should treat it as a private matter as well.
Although he preceded the Hasidic movement, the Ramchal, Rabbi Moses Chaim Luzzato supposedly had no beard. He did however write a section in his sefer Mesillat Yesharim dealing with Chassidut. I found a source (http://ha-historion.blogspot.com/2006/11/beards-in-jewish-tradition.html) that suggests he was influenced by the renaissance culture developing around Italy at the time.
