In September, http://mic-ro.com/metro/24h.html answered my question.* Thanks, guys!
Their list, and my comments:
Chicago, USA: Red and Blue lines
Copenhagen, Denmark: Metro
London, United Kingdom: Thameslink only (no tube lines)
New York, USA: 20 subway lines, SIR, PATH, partly LIRR
Philadelphia, USA: PATCO
Chicago's Red and Blue Lines are the busiest two lines in the system and cover a substantial part of the city. It's a pretty serious attempt to run a working service all night. Also, because the Blue Line serves O'Hare, it means you can get to the airport 24 hours a day. (Or leave it 24 hours a day, whichever.) No service to Midway, however.
The Copenhagen Metro opened in 2002 and features driverless trains, which makes overnight running less dependent on staff: an important consideration in high-wage Denmark. Its two lines are complemented by a much more extensive S-tog (suburban rail) network which does not run overnight. The Metro itself isn't a very big system but in concert with the S-tog, which is used like a metro, service covers a lot of the city, just not all of it overnight. The train cars were designed to accommodate the disabled, including light signals for the deaf to warn that the doors are closing.
When they say none of the tube runs in London, they mean it. Thameslink is a single mainline railway route run by First Capital Connect which runs in a series of north-south tunnels from St. Pancras to Blackfriars. Counting it as an 24-hour metro service is a bit of a stretch. It's as if the New York City subway closed at night but you counted LIRR service from Jamaica to Penn Station as overnight metro service. On the other hand I say technicalities count, since we only have five systems that actually run 24/7, so it's in, even if it only runs once every half hour during the overnight.
"20 subway lines, SIR, PATH, partly LIRR" covers the vast majority of New York's system. You can get to almost every station at any hour of the day or night. In fact, they're not even designed to close. Many entrances lack gates or doors, which makes securing them during strikes more challenging.
An additional point I'd add to New York's listing is that AirTrain JFK also runs all night, making it possible to get to Kennedy Airport any time as well. This may make Kennedy and O'Hare the world's only two airports with 24/7 rail service that actually goes somewhere beyond the immediate area of the airport. (Many airports have inter-terminal people movers that run all night. AirTrain Newark even takes you out to the New Jersey Transit Newark Airport rail station. Unfortunately, then you're stuck waiting on the platform until morning, because that station doesn't have any service that takes you away from the airport overnight.)
Until 1991, Philadelphia ran "Owl" service overnight on both the Broad Street and Market-Frankford lines, but it's been bustituted since then. It saves them money.
PATCO is the cross-Delaware equivalent of PATH. It still runs 24 hours, should you have some pressing need to visit Camden in the wee hours of the morning.
And that's it. Kudos to mic-ro.com for doing the research. Someone should edit the relevant Wikipedia articles.
Edit: I missed the obvious point that Copenhagen's Metro goes to Kastrup Airport, which makes it the third airport with a 24/7 rail link.
*I'm sure they've never actually seen my post or even heard of me.
I did specify 24/7 in my question, but the folks at mic-ro.com did find number of systems--Athens, Berlin, Bochum, Hamburg, Stockholm, Vienna, Warsaw--which run overnight Friday and Saturday, and two in Spain--Barcelona, Valencia--which run overnight Saturday only. See the page for details.