Q. Can one pick up a soiled tissue (Kleenex) paper from the floor on Shabbat or is it muktze?

A. Garbage is usually considered muktze since it has no use, as Shulchan Aruch (O.H. 308: 27) rules regarding inedible food leftovers. However when something is disgusting or is malodorous, to the point that it prevents ordinary use of the area around it, the restriction of muktze does not apply, and it is permitted to remove it. This is known as a geref shel re’i. (O.H. 308:35-37).
Even if the garbage has not reached that ominous stage, it may not be considered muktze, because it could be an item that is still usable. (see Shmiras Shabbos Kehilchasa 82,45).
In regards to a soiled tissue paper on the floor, that was used for cleaning someone’s nose, you may argue that some people would find it repellent to the point of considering it a geref shel re’i, but most just would not pick it up, even if it could still be used to wipe something else and it may become muktze.
However, Horav Shlomo Miller’s Shlit’a opinion is that since they would pick it up using a clean tissue to remove it or to clean a wet spot in the floor and similar if needed, it is not muktze.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a