The mishna in Shabbos 143a says that it is forbidden to squeeze the juice out from the fruit on Shabbos for the purpose of drinking the juice. To extract the juice from the fruit is forbidden by the Torah under the melocha of mefareik, which is a toldah of dosh. Tosafos in Kesuvos 6a says that to squeeze a towel which is absorbed with water in order to use the water, is forbidden under the same transgression.
According to this, it would be forbidden to use a sponge to do dishes. The reason is that water is absorbed into the sponge, and when scrubbing the dishes pressure will be put on the sponge to extract the water needed to clean the dishes.
A similar application to this rule is the use of baby wipes to clean a baby. In order for the wipes to work, pressure is placed while rubbing, causing the absorbed liquid to be expelled from the wipe. This is forbidden from the Torah just as squeezing the juice from a fruit. Instead, water or oil should be put on the baby first, followed by the use of a dry tissue.

Rabbi Tzvi Aryeh Hyman, Mir Yeshiva
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