R. Akiva Eiger in his comments on the Shulchan Aruch (302:Magen Avrohom 6) writes that one may prepare things on Yom Tov for Shabbos (provided that no melacha is being done) even without an eiruv tavshilin. Although Tosafos 113a writes that one may not fold their talis on shabbos for after shabbos, one may fold their tallis on yom tov which is erev shabbos for use on shabbos. R. Akiva Eiger explains where he derives this from. In the Mishna 113a there is a Machlokes between R. Yishmael and R. Akiva whether the left over korbanos of Y.T. can be burned on Shabbos. R. Yishmael holds that left over korbanos of Y.K. can be burned on Shabbos, but left over korbanos of Shabbos cannot be burned on Y.K. because shabbos has a higher kedusha than Y.K. The Mishna implies that it is for the same reason that R. Yishmael permits setting the beds on Y.K. for Shabbos, since Shabbos has more kedusha, but not the reverse. The rationale is that the prohibition to be טורח on Shabbos or Y.K. for afterward, doesn't apply when one is preparing on a day that has less kedusha for a day that has more kedusha. Whereas, R. Akiva holds that the kedusha of Shabbos is not greater than Y.K. so he forbids the left over korbanos from one to be burned on the other. Based on R. Akiva it would also be forbidden to do activities on Y.K. to prepare for Shabbos. Although we follow R. Akiva and forbid and preparation from Shabbos to Y.K., the Braisa 114a says that even R. Akiva admits that left over korbanos of Shabbos can be burned on Yom Tov, because there is no question that the kedusha of Shabbos is greater than Yom Tov. Therefore it should follow that one may make preparations on Yom Tov which falls on Friday to prepare for Shabbos. R. Akiva Eiger points out that the Eliyahu Rabba disagrees and holds that when Y.T. falls on a Friday, one can only do things from Y.T. to Shabbos if they have an eiruv tavshilin.
The M.B. 302:17 quotes this fundamental argument as to whether an eiruv tavshilin is necessary for preparations that don't involve melacha. R. Akiva Eiger would permit one to fold his talis on Y.T. for Shabbos, whereas the Eliyahu Rabba would only permit it if one had made an eiruv tavshilin. The Aruch Hashulchan 302:10-12 justifies the minhag to fold talisos even on Shabbos for after Shabbos based on Rashi and the Rambam who forbid מקפלין את הכלים ומציעין את המטות only because of a מתקן כלי issue since they were very meticulous about the creases, which doesn't apply to our talisos. He implies that acc. to Ra'avad 23:7 and Tosafos who imply that it is an issue of doing a טרחא for after shabbos, it would be assur. It seems to me that nowadays the folding is necessary to properly fit it into the bag so that the shul doesn't look messy and no one takes the talis, it isn't considered preparation for after shabbos and therefore permitted even according to Ra'avad and Tosafos.
Rashi 114a implies that according to R. Akiva we consider Shabbos and Y.K. to have equal levels of kedusha, which is why the left over korbanos of shabbos cannot be burned on Y.K. and visa versa. Also, that is why one cannot prepare from Y.K. to Shabbos, as the Mishna implies. However, in the braisa in Megilla 23a there is a machlokes R. Yishmael and R. Akiva regarding the number of aliyos on Shabbos and Yom Kippur. R. Yishmael holds 6 for Y.K. and 7 for Shabbos which is very in line with his approach here that the kedusha of Shabbos is greater. But R. Akiva holds 6 for Shabbos and 7 for Y.K. which implies that the kedusha of Y.K. is greater than Shabbos. Why then does R. Akiva hold that the left over korbanos of Y.K. cannot be burned on Shabbos, he should hold the exact opposite of R. Yishmael, that leftovers of Shabbos cannot be burned on Y.K. but leftovers of Y.K. can be burned on Shabbos? Also, why does Rashi write that according to R. Akiva the kedusha of Y.K. is like shabbos - רבי עקיבא היא דחשיב ליה יוה"כ כשבת, when in truth it is a higher level kedusha than Shabbos? The Turei Even in Megillah wonders what advantage Y.K. has over Shabbos practically that would justify and extra aliya, and he suggests that it is due to the extra tefillah of ne'ilah. However, all the rationales offered by the gemara seem to be simanim that indicate a higher level of kedusha, so the extra aliya for Y.K. should also indicate an extra level of kedusha over Shabbos - וחזרא קושיין לדוכתיה וצ"ע.