Karate teachers in the 1980's did the world a huge disservice by spreading the myth that you should not wash your belt because of some made-up ancient Asian traditions. In striking arts like karate or taekwondo, not washing a belt isn't that big of a deal because there's rarely close contact between the practitioners. However, in jiu-jitsu, everything is close contact, and an unwashed uniform (including the belt) will result in the spread of harmful bacteria like staphylococcus and mite infestations from one practitioner to another, from the uniforms to the mat, and from the mat to the athletes.
For everyone still perpetuating the myth that a dirty belt = ancient mystical samurai power, know this: belt ranks were first introduced by Jigoro Kano (the founder of judo), around the year 1900. Other modern "traditional" martial arts just copied the judo belt ranking concept. A hundred years might seem "ancient" to some people, but it's really not that long ago. That was my grandma's generation. The first black belts were actually black, they weren't just really dirty white belts stained with blood, sweat, and dirt from hard work.
Wash your belts kids. A dirty belt doesn't make you a mystical ninja, it just spreads staph infections. If you've never had one of those, trust me, you don't want one. Also, if your belt is brown, but you haven't earned a brown belt yet, it's time to toss that thing out and get a new one.
Your martial arts ability is not washed away when you toss your belt in the washing machine with your gi, but it might be if you get infected with flesh eating bacteria or scabies and end up looking like something from The Walking Dead.