I posted an answer for a recent question about a forest fire that started after your hike began, that said:
Presumably you registered at the trailhead, and you have a friend who knows where you are. There should be aircraft in the area soon attempting to identify your location and risk level. Source
A comment was posted in response
The trailhead registry will probably burn in a forest fire. source
In my experience the trailhead registry is usually elevated on posts in a mostly cleared area. The paper register form is inside a closed boxed, that should protect it from the rain and any fire.
I assume that in the event of a new forest fire, the forest rangers would check all the trailhead registries in the area to assess for hikers at risk.
Googling images found mostly what I expected. I searched and did not find any thing one way or the other about what the likelihood of the registry burning before the rangers had a chance to retrieve it.
NOTE: In many cases hiker must register at the ranger station, or through some other means (electronic). This question is only about paper registry at the trailhead location.