All perfect praise be to Allaah , the Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
It is not proven that the Prophet washed his hand or ordered the Companion to wash his hand due to touching the dung although he stated that it is impure. The reason could be the fact that purification in this case is not binding because the dust of an impure object is pardoned, according to some scholars.
As-Suyooti, a Shaafi‘i scholar, may Allaah have mercy upon him, stated that there are kinds of pardoned impurity. He said: "…and dust of a dry impure object…"
Al-Buhooti, may Allaah have mercy upon him, a Hanbali scholar, said: "The smoke, dust and steam of impure things are pardoned as long as there is no effect on the pure thing because it is difficult to keep away from such things. Some scholars said that they are pardoned as long as they are not dense."
Az-Zarkash, a Shaafi‘i scholar, may Allaah have mercy upon him, said: "There are different kinds of pardoned impurities; among them is the dust of a dry impure thing."
As-Suyooti, may Allaah have mercy upon him, also said: "If an impure thing touches a pure one while both are dry, the latter is not affected by the former."
Accordingly, washing the hands from the dust of the mentioned impurity is not binding since it falls under the pardoned type of impurities.
Allaah Knows best.