The Shulhan Aruch (302:3) presents restrictions regarding folding clothes on Shabbat. It should be noted that there was the opinion of the Kol Bo, one of the Rishonim (Early Authorities) that the prohibition of folding mentioned in the Talmud does not apply nowadays, as our clothing and folding are different. Clearly, Maran did not accept that opinion.
The Shulhan Aruch does permit folding only against the crease, as this has less of an effect on the garment. The Poskim disagree as to whether the leniency of folding against the crease is only if it is done L’Sorech Shabbat. This is the position of the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909). However, the Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933), based on the Elyah Rabba, understands that the leniency applies in all circumstances.
According to many opinions, including the Ben Ish Hai and Hacham Ovadia, one should not fold his Tallit on the crease. Other Poskim, including the Eres Haim, Poal Eliyahu and Shemesh U’Magen, state that the Minhag was to fold the Tallit on the crease. This was also the Minhag of Haleb. If someone has that custom, there is room to be lenient.
Hacham Ovadia rules that according to the stricter opinion, one may continue to fold against the crease, and not be concerned that he is forming a new crease, as long as the original crease is still recognizable. However, Shemirat Shabbat K’hilhata cites Rabbi Shlomo Zalman Aeurbach (Jerusalem, 1910-1995) that in order to avoid this problem, one should re-fold his Tallit on Mosa’eh Shabbat according to the original crease. However, Hacham Ovadia rejected this stringency.
One should not fold his Tallit along the original crease on Shabbat, but against it. It is not necessary to re-fold the Tallit on the crease on Mosa’eh Shabbat. Those who have the custom to fold the Tallit on the crease have what to rely on.