Many people have a filter mechanism attached to their faucets, such that the water is automatically filtered on its way through the faucet. Similarly, some people have special pitchers with a built-in filtering mechanism that filters the water on its way out of the pitcher. Is it permissible on Shabbat to use a faucet or pitcher that has such a mechanism?
This question is discussed in the work Shemirat Shabbat Ke’hilchatah (3:56), and he rules that using such an apparatus is allowed on Shabbat, provided that it had been attached to the faucet or pitcher before Shabbat. Since it is built in to the faucet, there is certainly no concern that one may squeeze the filter during Shabbat. Shemirat Shabbat Ke’hilchatah adds that if the filter falls off the faucet on Shabbat, one may return it on a temporary basis, without screwing it in fully for permanent use, as this would constitute "Boneh" ("building," which is forbidden on Shabbat).
There is, however, one condition that must be met for a filter to be permissible for use on Shabbat (as Shemirat Shabbat Ke’hilchatah writes in Halacha 49). Namely, the water must be drinkable even without filter. If the water would otherwise be soiled and unsuitable for drinking, then one may not use a faucet or pitcher that automatically filters the water and renders it drinkable. Halacha allows using an automatic filtering system on Shabbat only if the water is suitable for drinking even without the filter, and the individual would drink the water unfiltered, but he uses the filter so that the water would be especially clean. Here in the New York area, water is generally drinkable from the tap even without filtering, and therefore it would be permissible to use a faucet or pitcher with a built-in filtering system on Shabbat.
Summary: It is permissible on Shabbat to use a faucet or pitcher with a built-in filtering mechanism, assuming that the water is basically drinkable even without filtering.