What is the ruling on cleaning the binding of the Mushaf with water and a tissue to remove dirt that cannot be removed except by this method?

There is no harm in cleaning the cover of the Mushaf with pure water, especially if cleaning depends on it. This might even be obligatory to glorify the sanctity of Allah and His rituals. Removing dirt from the Mushaf is obligatory, even if the dirt itself is pure, and it is not permissible to leave it on the Mushaf, because leaving the Mushaf in dirt is like placing it in dirt. Therefore, the Mushaf must be maintained and cleaned immediately if it becomes exposed to dirt.