The Shulchan Aruch writes (320:20) that if on Shabbat a person partakes of strawberries or other foods that leave a colored residue, he may not clean his face with a cloth, as he thereby "colors" the cloth, in violation of the Shabbat prohibition of Tzovei'a (dyeing). Halacha does not forbid eating such foods, despite the "coloring" of one's face and lips that often results, but the coloring of the cloth would, indeed, be included under this prohibition.
The Radbaz (Rabbi David Ben Zimra, Egypt, 1480-1574) in Helek 4, Siman 1201) disagrees, claiming that wiping one's face has the effect of soiling the cloth, and this therefore does not constitute dyeing. He adds that he never saw anybody make a point of not cleaning his mouth after eating strawberries and the like on Shabbat.
Nevertheless, since the Shulchan Aruch rules stringently in this regard, a person who wishes to clean his face on Shabbat after eating foods that leave a colored residue should first wash his face with water and only then wipe it with a cloth. If no water is available, then he may clean his face with a cloth even without water.
It should be noted that this Halacha applies only to cloth napkins; one may use disposable, paper napkins to clean his face on Shabbat, and he is not required to first wash his face with water.
Summary: If one eats foods such as strawberries that leave colored residue on his face, he may clean his face with a disposable napkin. He should not clean his face with a cloth napkin until he first removes the residue with water.
(Halichot Olam, Helek 4, Page 284)
