מטפחות הַסְּפָרִים - The תפארת ישראל says that this includes a mantel, a cloth used to cover a ספר תורה, tablecloths used in a shul, or even tablecloths used for dining.

רבי אליעזר אוסר - The הלכה follows R' Eliezer.

מטפחות הַסַּפָּרִים - The רע"ב and רמב"ם says that this is referring to barbers' smocks that have a neck-hole, like a poncho; implying that a simple sheet, draped over a person to keep themselves clean while at the barber, would not pose a problem.  However תוס' רעק"א asks why this would be permitted, as even placing כלאים on one's body is forbidden, even if it is not דרך מלבוש, just as is the case in the next משנה regarding the prohibition against placing a packsaddle of שעטנז on one's shoulder to take out garbage. (This is called העלאה, per ויקרא יט:יט which reads ובגד כלאים שעטנז לא יעלה עליך, which adds an additional stringency beyond that given in דברים כב:יא, which reads לא תלבש שעטנז.) The תפארת ישראל speculates that perhaps when the שעטנז is used to protect one body (as by the saddlepack used to protect one from garbage), then העלאה is prohibited, while when used merely to protect one's clothing (as is the case by a barbers' smock), only when done בדרך מלבוש is it not allowed.

ומרדעת של חמור - This is not as soft as the cushions and mats in משנה ב, and so even if his skin touches them they are permitted.