Is it permissible to fold one’s Tallit on Shabbat, and, if so, may he fold it regularly, or must he fold it differently than the way he normally does?
Tosafot (Talmud commentaries from Medieval French and German scholars), in Masechet Shabbat (113), write explicitly that folding a Tallit is forbidden on Shabbat, since it is done in preparation for the following day. When one folds his Tallit after the prayer service on Shabbat, he obviously does not plan on wearing it again that day. As such, the folding is considered an act of preparation for the following day, which Halacha forbids on Shabbat.
The Shulhan Aruch (Orah Haim 302), however, approvingly cites the view that allows folding a Tallit on Shabbat in a different manner from the way one normally folds it. Meaning, according to the Shulhan Aruch, it is permissible to fold one’s Tallit provided that he does not fold it along the creases.
Importantly, however, Rav Haim Sittihon of Halab, in his work Eretz Ha’haim, records that the practice in Halab was to fold one’s Tallit on Shabbat in the normal fashion. It appears that this was the custom in Halab since even before the time of the Shulhan Aruch, and we generally assume that we may continue following a custom that dates back to before the times of the Shulhan Aruch, even if the Shulhan Aruch rules otherwise. Therefore, people who fold their Tallit in the normal fashion on Shabbat may continue doing so. This is especially so for those who have a special Tallit designated for use on Shabbat, in which case they fold not in preparation for the weekday, but rather for the next Shabbat, which is permissible.
Summary: There is a custom that permits to fold one’s Tallit on Shabbat, even in the usual manner, along the creases.