I was going to just let to go till this morning, but when I discovered that the drip would fill a 9x13 cake pan to overflowing in little more than an hour, I decided to give the emergency maintenance number a call. Turns out she couldn't really do much more than talk me through attempting to close the shutoff valve -- unsuccessfully, because it's kinda broken -- (I mean, she would have come over and done it, but I own tools and I was there dealing with it already, so it was easier to just do it myself) and send the landlord over this morning.
So I emptied a number of pans of tepid water last night and this morning, and he came over and tinkered with it. The "zone valve" (whatever that is) was leaking, and he fixed that, so it's not leaking any more, but the shutoff valve is also very broken, and now that's leaking a tiny bit.
Which means that Monday (when the real plumber guy is available), they'll have to shut off the heat to the whole building and go into all the apartments to drain the water so they can replace both (very broken) shutoff valves in our apartment. (Thankfully, it's plenty warm and our apartment is well insulated.)
There are times when I really appreciate living in a place where maintenance issues are someone else's problem...