His work in washing dishes is permissible in itself; in fact, it is part of the recommended cleanliness. The fact that he is washing dishes from prohibited or impure items does not matter, similar to a tanner who deals with the hide of a dead animal which is impure before it is tanned.
As for the dishes being used again, this goes beyond his responsibility and he is not accountable for it. The items could be used for various purposes, similar to a knife seller; the buyer can use the knife for something permissible or something prohibited, and the responsibility lies with the user, not the manufacturer.
Washing dishes is part of the recommended cleanliness, so there is no harm for those who perform this job.
Fatwā issued by Dr. Khālid Naṣr