The Sages tell us that Ezra Ha'sofer (Jewish leader during the period of the founding of the Second Commonwealth) enacted a prohibition against washing clothes on Erev Shabbat, so as to ensure that the day would be devoted to making the other preparations necessary for Shabbat. Accordingly, the Ben Ish Chai (Rabbi Yosef Chayim of Baghdad, 1833-1909) in Parashat Lech-Lecha, Halacha 8, wrote that one should make a point of washing clothes for Shabbat on Wednesday or Thursday.
Does this prohibition apply today? In ancient times, washing clothes entailed going to a river, washing all the clothes by hand, and hanging them out to dry. This was a process that in some cases could take nearly half a day. Today, we simply throw the clothes into a machine, add some detergent and press the button. Given the relative ease and speed of laundry nowadays, would it be permissible to do laundry on Erev Shabbat?
Chacham Ovadia Yosef, in Halichot Olam, Helek 3, page 50, ruled that nowadays one may, indeed, do laundry on Erev Shabbat, since the process of washing clothes today is far less arduous and time consuming as it was in ancient times, and hence does not detract from one's other Shabbat preparations.
Is it permissible to put clothing in a washer or dryer just before Shabbat and allow the machine to run on Shabbat?
According to the practice of the Sepharadim, it is permissible to turn on a machine before Shabbat and have it run on Shabbat, whereas the Ashkenazim are stringent in this regard, based on a ruling of the Rama (Rabbi Moshe Isserles, Poland, 1525-1572), in his glosses to the Shulchan Aruch (252:5). (See Yehaave Daat, Helek 3, siman 18.)
Summary: It is permissible to wash and dry clothes in washing and drying machines on Erev Shabbat, and Sephardic custom allows putting clothes into a washer or dryer before Shabbat and having it run on Shabbat.