My guess - and please note it is a guess! - is that the heat of the drying cycle will gas off any chlorine. If the items are dry by the time you use them, I don't think you will have any chlorine issues.
That said ... for me the easy process is to put all of my utensils into my large cheesemaking pot with two or three inches of filtered water, and bring to boil. Boil 10 minutes. Remove to a clean cloth on a sanitized surface. (Yes, I've wondered whether the cloth could introduce problems - but none so far over several years of making cheese.) Note that I usually do not sanitize the colander, mold, plastic mesh, and cheese cloth at this point; instead, I wait until the make is 30-60 minutes from being ready to use these, and sanitize them with a separate pot of boiling water. The colander does not fit in the pot, so I "steam" it over the pot. Seems to work for me ...
While the above is happening, I pre-heat the milk by putting the jugs in the sink, and filling the sink with hot water. This saves significantly reduces the time for getting the milk up to the target temperature, so it is win-win for me.