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.