Update:

The third-party toner seemed to work fine, but the printer itself is decidedly not fine. It gives the red light, like it's jammed or something. There is clearly no paper anywhere around, so no jam to fix. Power cycling or whatever just causes the same behavior. One of the motors tries to turn for about a second, then *clunk* and it's got the red light back on again.

Is it dead, Jim?

EDIT: So I found the service manual from some dodgy Russian web site. The service manual helpfully mentions there are menus that might tell you service related things, e.g., "Menu-Setup - Maintenance - Check other - (Select a desired part)". Unfortunately, there's no such menu available anywhere from the web interface of the printer. All I know is "Stop Printing, Service Requested". I'll also note, for the record, that I've found at least one other bloke with the same exact problem.