From the blog of R’ Shlomo Aviner, who advocates studying Torah on Nittel Nacht:
“Ha-Rav Moshe Sternbuch [currently on Eidah Charedis] wrote that this custom [of not learning on Nittel Nacht] 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).”
However, the Lubavitcher Rebbe always observed this custom. His practice was apparently to play chess.
Anyway, if R’ Sternbuch is correct that this custom was unknown in Lita, then why should today’s Litvishers follow it? The Chasam Sofer followed it but he was, um, Austro-Hungarian.