How long can a HPS ignitor be left with a dead bulb? I'm worried about finding a replacement ignitor, and it's the bare circuit board type.

is the bulb completely dead, or does it start up and then go out?  Some are "smart" type that will not attempt re-ignite for a certain amount of time or next power cycle. 

Depends on design. Some will die in a couple weeks (but this means, that even before that they are allready damaged i.e. their reliability is compromised even if you replace the lamp when the ignitor still works, and it may fail later) and some can go on clicking for 12+ years with a missing lamp

Most will go at least 90 days with out any issues, but I've seen them last years..