Assuming it's not the nine days, is there a reason why one should not do laundry on Motzei Shabbos?
Recently I've been told by someone that he doesn't.
Nahalat Avot page 74
"We are accustomed to keep away from different work and responsibilities on Mosae Shabbat." This doesn't directly answer, but it brings down an opinion about work on MS. By the way, this book has Haskamot from Rav Ovadia Shelita, Rav Mordechai Elyahu, Rav Bakshi Doron Shelita, and Rav Amar Shelita.
I'm guessing it has something to do with the idea that Motzei Shabbat is something of a 'minor holiday' (there are many people who have a special meal on Motzei Shabbat for this reason), and for that reason some people have a minhag not to do laundry then (similar to a woman not doing any housework for a half hour after lighting channukah candles).