Question: Is it permissible to learn Torah on "Nittel Nacht" (Christmas Eve)?
Answer: There is a custom among some Chasidim not to learn Torah on "Nittel Nacht" in order not to contribute positive spiritual powers to idolatry. The Lubavitcher Rebbe related in a talk in the name of his father-in-law, the Rebbe Rayatz, that the 5th Lubavitcher Rebbe, the Rebbe Rashab, played chess on "Nittel Nacht" (Parashat Vayeshev, 1st Night of Chanukah 5750). And this was indeed the custom of the Lubavitcher Rebbe until midnight of "Nittel Nacht" (see Sha'arei Halachah U-Minhag vol 3 pp. 64-67. There is a famous picture of the Lubavitcher Rebbe and his father-in-law, the Friediker Rebbe, playing chess on Nittel Nacht). But our custom is that one needs to learn Torah on that night, just like every night. Some explain that the reason for not learning Torah of that night is that the Christians would become impassioned on their holiday, would go out and kill Jews (Ta'amei Ha-Minhagim p. 500). We therefore would need to stand guard instead of learning Torah (Ta'amei Ha-Minhagim p. 500). Baruch Hashem, Christians do not act this way today and we need to learn a lot of Torah.
By the way, it is told in the book "Siach Sarfei Kodesh" (p. 192) that when the last Lubavitcher Rebbe was sitting shiva, the Satmar Rebbe, Ha-Rav Yoel Teitelbaum, paid a shiva call and asked him: Why do we observe the night of "Nittel Nacht" according to the non-Jewish calendar (which is forbidden) instead of according to the Jewish calendar? The Lubavitcher Rebbe answered that we do not learn on that night in order not to give spiritual strength to the god of the Christians and we therefore observe it on the night that they celebrate, which is according to their calendar. And – he added – we must therefore be aware when they celebrate this holiday in the place we are located. In most places, their holiday is on December 25th, but it other places it is on a different day (For example, January 6th). In such places, that is the night of "Nittel Nacht".
But as we said, our custom is to learn a lot of Torah on that night. And Ha-Rav Moshe Sternbuch wrote that this custom was unknown in Lithuania and it is only a custom among Chasidim (Shut Teshuvot Ve-Hanhagot 1:551). The Chazon Ish would learn on "Nittel Nacht," and said that it was forbidden to waste time from learning on this night and he criticized those who did not learn on that night. The Steipler Gaon would also learn on "Nittel Nacht," but did so by heart so as not to upset those who have the custom not to learn. The Steipler Gaon also requested not to be informed when Nittel Nacht is so that he would not have to waste time from his learning (Orchot Rabbenu vol. 1 p. 193). And Ha-Rav Ovadiah Yosef has written that no such custom exists among Sefardic Jews (Shut Yabia Omer vol. 7 Yoreh Deah #20). The Lubavitcher Rebbe also writes that this custom does not exist in Eretz Yisrael (Sha'arei Halachah U-Minhag ibid. p. 67), even though it seems that some do have this custom in Eretz Israel.