Question:
May one fold his tallis on Shabbos after davening?
Answer:
If one plans to wear the tallis again later that same Shabbos, then it is permitted to fold it in its regular manner. If, however, one does not plan to wear it again until after Shabbos, then it is forbidden to fold it in its normal fold on Shabbos. Nevertheless, it is permitted to fold the tallis in a different manner than its usual fold, even if one does not plan to use it again until after Shabbos. However, some Poskim add that after Shabbos one should then refold it in its regular manner. Practically, many meticulous Jews are accustomed not to fold their tallis at all on Shabbos, even in an irregular fashion. Likewise, there is a custom to fold the tallis after Shabbos.
Regarding a borrowed tallis, one is not required to fold it on Shabbos, even if it was originally found folded.
Sources:
Admur 302:8-9; Ketzos HaShulchan 117:2 and footnote 8; Daas Torah 302:3; Nemukei Orach Chaim 302:1; Shemiras Shabbos Kehilchasah 15:46; Admur 300:4.