If freight elevators count, DePaul's Lewis Building has (had?) a manually-operated open cab that looked original. Also, the former Douglas plant/USAF building #4 at O'Hare, (two stories, visible from Higgins and Mannheim), has one that dates to 1942.
Serhii, I was in this building at Ravenswood & Waveland a couple of weeks ago and took a ride on a VERY cool old freight elevator with old hardwood floors and the gates that you had to manually close. It looked very old to me so I asked about it and the guy I went to see told me it was from 1935.
My freight elevator is original and 'manually' operated, i.e. you have to hold the up or down button to take you to your floor as well as summoning it. Building was built in late 20's. I was in another 20's apartment building which still had all the original deco accoutrements in the elevator, gold leaf, check, open grill doors, check... There are a lot of smaller residential buildings with 20's elevators which have been sensitively upgraded, renovated or restored with updated mechanicals.
The Powhattan (50th and S. South Shore Drive) still have elevator operators for it's passenger elevators, but it's a luxury building.
Technically not in Chicago but...
I sometimes work in downtown Hammond and the building at 5246 Hohman Avenue still has both a working old freight elevator and a passenger elevator which is operated by a very nice lady named Sheila during regular business hours. If you are ever in the area it's worth a visit.
In nearby Oak Park (where I live) the Write Inn has an original elevator from I think 1898. They keep it maintained, and there's a little sign about it being the "oldest elevator in Oak Park."
Also, I remember about 5-6 years ago, when I first moved here... A friend of a friend was dating some local rapper and took a couple of us to her boyfriend's "studio" on the Westside. Not sure exactly where, but I think it was on California Ave. in Humboldt Park. The building was an old Louis Sullivan warehouse, and had a working freight elevator. Riding it was scary at hell... Almost as scary as being in a 100 year old warehouse in the middle of the Westside at 11:00 at night!
I don't know if this counts because the buildings hve been demolished, but the building, actually twin buildings, we lived in during the late 40's and early 50's, the Hyde Park Towers at 810 and 818 Hyde Park Blvd had 2 elevators in each building. Since the building was built as accommodations for the Colomian Exposition, I assume that the elevators dated from that time. The 2 passenger elevators in the front were old and rickety and were prone to getting stuck occasionally. The freight elevators in the rear were wonderful contraptions with manual cages, hardwood floors, open overheads and ropes that operated them. I used to ride the with the janitor or building engineer, but never figured out how to work them, of cours I was onl 6 to12 years old then.