Can one use a dishwasher to wash both meat and dairy dishes, either simultaneously or consecutively? What about pareve dishes?
Rav Ovadia Yosef permits meat and milk dishes at the same time if there is soap in the initial wash, but prefers that one only run them in separate cycles, as, he reports, does his son Rav Yitzchak Yosef (Yabia Omer YD 10:4).
Rav Yitzchak Abadi permits washing meat and milk dishes in separate cycles (Or Yitzchak YD 1:4 and 2:8).
Rav Shlomo Aviner permits washing meat and milk dishes in separate cycles (Sheilat Shlomo 1:311, 2:174, 3:135 quoted here).
Rav Dov Lior permits washing meat and milk dishes in separate cycles if you make sure the dishwasher is clean between runs (http://www.yeshiva.org.il/ask/60607).
Rav Yosef Kapach permits washing meat and milk dishes in separate cycles (quoted here http://www.tora.co.il/shiurim/tsuriel/p_kapach.doc).
It is my understanding that most of the Rashei Yeshiva in YU (including Rs Willig and Schachter) believe running an empty cycle between a meat load and a dairy load would be more than sufficient qua Kashrut (though whether routinely switching is good kitchen strategy is another discussion).
Badei Hashulchan (Beiruim 95 sv Shari) rules that if it has been 24 hours since the last use and the dishwasher (including the filter at the bottom) is cleaned of any residue, then one could be lenient, but he doesn't think such a cleaning of the filter is very practical and cautions against trying. (Anyone who has ever tried eating the guk from the filter may at this point be baffled.)
Igros Mosheh Yoreh Deah Orach Chaim 1 siman 104 & Yoreh Deah 2 siman 28 & 29 says you can use a dishwasher for both meat and dairy immediately one after another, but only when you have reserved racks, one for meat, and one for dairy.
He does say that there is room for a baal nefesh to be stringent and not rely on this.
Chacham Ovadia Yoseph has a very important Teshuba (answer) granting permission for this in his book ‘Yabia Omer’ in Helek 10. There, he writes an opinion based on what Maran wrote in the Shulchan Aruch, in Yore De’Ah siman 95:4. Maran writes of the case where people wanted to use giant pots normally used for dairy cooking for the purpose of cleaning various tableware and cookware including meat dishes. Maran says that it would be permissible to use this giant dairy pot to clean meat dishes so long as a distasting agent such as ‘Efer’ (Ashes) or soap is added. Such an agent would render the milk in the walls of the pot ‘pagum’ (bitter, unsavory, unfit, spoiled), thus allowing the insertion of a meat vessel for cleansing. This leniency even applies if the meat pot is dipped in the giant milk pot together with boiling hot water.
Chacham Ovadia Yoseph applies Maran’s reasoning by the giant milk pot to permit simultaneous washing of meat dishes and dairy dishes in a dishwasher. The soap deposited into that designated compartment when released together with the hot water renders the dishwasher cavity ‘pagum’. Chacham Ovadia Yoseph therefore comes out that not only is it permissible to use the same dishwasher for both meat and dairy dishes, but it is also permissible for one dishwasher to be used to wash dairy and meat dishes simultaneously. This is also the opinion of Chacham Matloub Abadi in his Sefer ‘Magen Ba'adi’.