DVC is Disney's version of a timeshare. A timeshare it totally different from a hotel. You don't get daily service unless you pay extra for it. Thus, members take their own trash out if it gets too much before the fourth day when housekeeping comes to remove it.
Still, at a non-DVC resort, your best bet to resolve a problem is talk to someone who knows. For trash removal, call housekeeping and ask where you should put a soiled diaper if you don't want to leave it in your room.
Actually, if I had a smaller child still wearing a diaper, I would plan on disposing those in the trash can in my room. However, unless my child never had a blast from the past (which I would find very unlikely), I'd plan a head with some special deodorizer or disposal bag for my hotel room. That's why I recommended bringing the Diaper Genie. Never had a smelly room with my Diaper Genie way back when. Anyone know if they have come up with a travel version of the diaper genie yet? They'd make a fortune on that idea.