Q. Are crumbs Muktza after Bentching? What about matzah or cake crumbs?

A. Shulchan Aruch (308: 27) rules that even crumbs less than a Kezais size one is permitted to remove from the table since they are used for animal feed. In our days some save them to be fed to the free backyard birds after Shabbos. See Mishna Berura (110).
Chut Hashoni (3: 60: 1) is of the opinion that during the meal one may touch bread crumbs since it is the way of common eating.

Piskei Teshuvos (308: n. 339) quotes Orchos Rabenu that Horav Yaakov Yisrael Kanievsky Zt’l was careful following the Mishna Berura’s opinion not to remove the crumbs remaining on the table after the meal directly, but rather to use another item (such as a piece of cardboard).
Horav Shlomo Miller’s Shlit’a opinion is similar.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller, Horav Dovid Pam, Horav Aharon Miller, Horav Chanoch Ehrentreu and Horav Kalman Ochs Shlit’a