The Shulhan Aruch (Orah Haim 337:2) writes that it is forbidden on Shabbat to sweep the ground – meaning, earth which is not covered by tiling – as this has the effect of filling in holes with earth, which violates one of the Shabbat prohibitions. However, the Shulhan Aruch adds, it is permissible to sweep a tiled floor, as there is obviously no concern of filling holes with earth, and the Sages did not enact a decree forbidding sweeping a tiled floor as a safeguard against sweeping the open ground.
However, in the very next passage, the Shulhan Aruch speaks about washing floors with water, and in this context, he indeed forbids washing even tiled floors as a safeguard against washing untiled ground. In other words, although the Shulhan Aruch permits sweeping tiled floors on Shabbat, he forbids washing tiled floors with water.
The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) explains that sweeping is necessary for maintaining basic cleanliness, and so the Sages did not forbid sweeping tiled floors as a safeguard against sweeping untiled floors. Washing the floor with water, however, is not as vital, and therefore the Sages enacted a decree forbidding washing with water.
Hacham Bension Abba Shaul (Israel, 1924-1998) ruled that nowadays, when all our floors are tiled, and nobody has floors in the home with exposed earth, this decree no longer applies, as there is no concern that one might wash the ground in violation of Shabbat. Moreover, Hacham Bension adds, in certain places, washing the floor with water is critically important for sanitation, such as in hospitals. Hacham Bension thus ruled that when there is a need to wash a floor on Shabbat, such as if part of the floor became very dirty, one may clean the floor with a rubber squeegee. (Washing with a mop is not permitted on Shabbat, as this will invariably result in Sehita – squeezing water from the mop.) Although one should refrain from washing floors on Shabbat when possible, this is allowed if the need arises. This is also the ruling of Hacham Ovadia Yosef, in Hazon Ovadia – Shabbat (vol. 4), where he adds that if water spilled on the floor, one may use a rubber squeegee to move the water away.
Hacham Ovadia further notes that it is entirely permissible to ask a non-Jew to clean floors on Shabbat. Even if the non-Jew will use a mop, instead of a squeegee, to clean the floors, and will squeeze water out of the mop, this is still allowed, since the non-Jew uses a mop for his own convenience, and not for the Jew. Especially in a synagogue, where washing the floors is necessary for the sake of a Misva, a non-Jewish custodian may be asked to clean the floors.
Summary: It is permissible to sweep a tiled floor on Shabbat, and, when necessary, one may wash a tiled floor with water, using a rubber squeegee (as opposed to a mop). One may ask a non-Jew to wash the floor on Shabbat, even if he knows that the non-Jew will use a mop.