It is forbidden to dip a cloth into water on Shabbat, such as to clean a stain. Doing so violates the prohibition of Melaben – whitening a garment – and could likely also violate the prohibition of Sehita – squeezing – given the possibility that absorbed water will be squeezed out of the cloth.
However, Rav Moshe Feinstein (Russia-New York, 1895-1986), in his Iggerot Moshe (2:70), writes that the prohibition of Melaben does not apply to disposable cloths, such as napkins or paper towels. Since the napkin will be discarded, this does not fall under the Torah prohibition of laundering a garment. Therefore, one may moisten a napkin or paper towel – something that is not made from fabric – on Shabbat and use it to clean, as long as he ensures to wipe lightly so that no moisture is squeezed from the napkin.
Similarly, Rav Eliezer Waldenberg (Israel, 1915-2006), in his Sitz Eliezer (13:83, 16:9), writes that one may use moistened toilet paper on Shabbat, as long as he wipes gently to ensure that no moisture is extracted from the paper (and, of course, that the paper does not need to be ripped). As long as one avoids squeezing, he may use pre-moistened wipes, or moisten toilet paper and then use it, since it is disposable and thus not subject to the prohibition of Melaben.
Summary: One may not moisten a cloth made from fabric on Shabbat, but one may moisten a napkin, a paper towel or toilet paper and use it to clean, as long as he wipes gently to ensure not to extract absorbed moisture.
