I've heard people say dropping off the dry cleaning to mean take the clothes that need dry cleaning to a dry cleaner's. Can I use drop off the laundry to mean taking the laundry to staffed laundries? If not, what is the common way to refer to this household chore?
Yes, you can "drop off" almost anything, if it involves going someplace and leaving something.
I'm going to drop off the kids at school.
I need to drop off my car at the mechanic. Something's wrong with the engine.
Let me first drop off these reports at the office, then I'll join you for happy hour.
The challenge is context. As Jeff Zeitlin points out, it should make sense to the listener -- for example, here in California, I've never really heard of anyone who would "drop off laundry" that wasn't dry cleaning. Either you do it yourself at a laundromat, or you have your own machines. It certainly makes sense, but it's unusual enough that I'd have to think about what you mean.
The exception is at hotels that offer laundry services, but in this case you wouldn't need to drop off the laundry, but rather have housekeeping come by to pick it up from your room.
Hi, this is Room 404. Could you send housekeeping by to pick up my laundry? Thanks.