I get this question all of the time. The answer is that it depends on how you use your laundry room. If it is a pretty tidy room which has a washer and dryer, then you would put a Mezuzah.
If your laundry room is generally piled up with loads of dirty, smelly laundry. Then you would not put a Mezuzah. This is because it would be considered disrespectful to put a Mezuzah there. The same law applies to a bathroom.
Of course, we are assuming that your laundry room has the minimum 36 square feet that any room needs to be obligated in a Mezuzah. We are also assuming that the laundry room has some sort of door frame. If it doesn’t meet both of these requirements, it is not obligated no matter how immaculate it might be.
Let us say that a laundry room appears to require a mezuzah on account of its square footage and door frame, but there still remains the issue of the unpleasantness of the dirty and, perhaps, smelly laundry that routinely takes up space in the area. Since the recurring responsibility of someone visiting the laundry room is to continually restore the environment, by transforming all that nasty laundry back into clean, healthful, and fresh-smelling laundry, should the necessarily positive, though transitory, outcome for the laundry room and its contents overrule the unfavorable, though similarly transitory, conditions that usually precede the good results?