<p>From what I've seen, you can take a typical extruder heater, apply the designated supply voltage without temperature control, and as long as the heater isn't contacting something with a flashpoint below the temperature the heater reaches, the heater will not catch on fire.  Thus, unless one catches the filament on fire, it seems that thermal runaway of the extruder heater wouldn't normally start a fire.</p>
<p>I'm not sure what would happen if someone installed wrong components, such as a 12V heater to a 24V supply. What are situations that could cause a 3D print to catch fire?</p>

A normal hotend will not melt or cause fires, usually, see first video. However, if the power regulating unit fails as well and higher voltage is supplied (19 V on 12 V cartridge are enough) it can happen, see second video.
Fires are more likely caused by overheating wires, especially where joints are presents.
Hotend runaway does not cause fires from the hotend itself, but it could cause fire related to its wiring. Also, the uncontrolled temperature may make the hot end fan fail, worsening the situation.
Issues canbe caused also by a runaway on the bed and overheating MOSFETs.