Some people feel you should never wash a Karate belt (Obi), especially a "yūdansha" (black belt grade). This thought is do with the Karate tradition, honour, respect, dedication to achieve the belt, however this will be a matter of personal preference and hygiene.
Belts should not need washing as often as the karategi. Read more on how to wash karategi. If handled with care (not thrown on the floor) it should not get all that dirty, or sweaty if taken off straight after an intense session. But after a period of time if it may need washing at some point.
Carefully wash your belt (Obi) by hand and dry it naturally. Very similar to the karategi’s belts can shrink also. Washing machines and definitely tumble dryers should be avoided. Extra care should be taken with silk and satin belts.