The furnace is a Burnham Alpine 150 (ALP150BW-4T02) that we installed in Fall 2015, not long after we bought the house. No issues before this. It had a diagnostic ("Limit String Status") where it explained why it was refusing to start:

This initially looks like two faults: one with the Condensate Float Switch (ours doesn't have Thermal Link or Burner Door sensors) and then one with the Gas Press Switch / Low Water Cutoff / Flow Switch / External Hi Limit. But checking the manual it says:

Interlock input limit items are wired in series. Items down stream of the open limit will also appear as "open" (measure zero volts). When the first item in the string is resolved (repaired, or fixed) the down stream items will also be fixed.

There's also a diagram to illustrate:

This tells us there is one thing wrong, and it is in the condensate system, right? Unfortunately, no. After getting a plumber out, it turned out that the problem was actually with the previous sensor, the Auto Reset Hi Limit. I'm not sure how a sensor fails in a way that the fault appeared to be at the next sensor in the string, but that's apparently what happened.

While I'm frustrated with the diagnostic, I'm glad we have heat again!