The Shulhan Aruch (311:8) introduces the concept in Hilchot Mukse of "Tiltul Min Ha'sad"-indirectly moving a Mukse item. He brings the classic example of pulling a radish out the ground before it takes root. In this action, it is permitted to move the Mukse dirt indirectly through pulling the non-Mukse radish. However, Maran restricts this principle to cases in which the indirect movement of the Mukse is for the purpose of the non-Mukse item, in this case for the radish.
The Poskim discuss common applications of this principle, for example, if pistachio shells or olive pits (which are clearly Mukse, since they are not edible even to animals) were left on the Shabbat table. If there was a substantial amount that would constitute a "Graf Shel Re'i"- a repulsive heap, it is permissible to remove the Mukse directly. However, if it was not such an amount, it is prohibited to directly move the shells or pits.
The question is whether one may remove them indirectly using a permitted object such a knife to scrape them off the table into the garbage pail. Hagaon Rabbi Zalman (Rabbi Schneur Zalman of Liadi, 1745-1812, Russia) holds that using a knife is not considered "Tiltul Min Ha'sad"-indirect handling; it is considered direct handling. Therefore, it is prohibited. This also the opinion of the Ben Ish Hai (Rav Yosef Haim Baghdad, 1833-1909) in the second year, Mikes, where he says that it is prohibited to clean bones off the table neither with his hand or with a utensil.
However, on the other hand the majority of the Poskim are lenient, including the Ramban (Milhamot Hashem, Shabbat 48b), the Rosh (4:10). This is how the Taz (Rabbi David Segal, Poland, 1586-1667) and the Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) rule. Hacham Bension and Menuhat Ahaba also clearly permits using a knife to scrape off refuse from the table, and this is the accepted ruling.
Even if a knife is considered indirect, the question is whether a utensil designed for scraping and cleaning, such a broom, is also considered indirect, or is that considered the direct way of handling the Mukse. The Poskim are lenient and permit using a broom and a dustpan to sweep up Mukse items, for the purpose of cleaning the floor or table. Likewise, it is permitted to roll up a disposable tablecloth with all the Mukse refuse and remove it from the table for disposal.
It is permitted to clean Mukse items off a table or floor indirectly, using a knife, broom or dustpan.