If there is a general purpose room (like a bedroom) wherein a child's soiled diapers are often disposed of, does that room take on the status of tashmish she'aino shel kavod - a room who's use is not dignified (Yoreh Deah 286:4), and therefore becomes exempt from mezuzah?
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The Ben Ish Chai sefer 2 Parshas Ki Tovah halacha 20 says that if one goes in and out and uses the place like a cattle room (as long as it is not disgusting) then it has a status of a room. A room where women wash themselves it would seem they don't need a mezuzah but if there is a door and it is locked during use then one is obligated in mezuzah from the outside.
A bathroom, or bathhouse, tannery, or mikvah are not a dwelling of honor and are patur (exempt).
It seems that there is an in-between type case which if one uses the place in an kovadik (honorable) way sometimes it is fine, but if it is always dirty then it is patur.
This is all speculation, please read it inside. I brought this as source because it seems to answer your question, but it can be debated.