This claim is entirely false, the Hollywood sign did not catch fire during the recent LA fires. The initial post linked to the claim has been taken down, so I looked for other sources.
With a quick google search, several sources disprove the claim. According to ABC 10 News San Diego, the sign was close to the fires but did not catch whatsoever. They announce that all images circulating of the sign on fire are fake.
Next was an article from the Los Angeles Times, titled “L.A. firestorms bring waves of fire myths, disinformation. Here’s how to debunk it and not get fooled.” Writer Karen Garcia claims that an AI-generated photo had been spread around the internet. This article provided its own fact check, with confirmation from Jeff Zarrinnam, the chairman of the Hollywood Sign Trust that the sign is untouched by the fires.
To further disprove this claim, I found an article from CBC Kids News titled “Watch – The Hollywood sign didn’t burn. How to spot fake L.A. wildfire posts.” This source also cites the Hollywood Sign Trust as their source as well as sharing tips on how to spot fake content.
I was able to find information about the Hollywood Sign Trust on the Hollywood Sign website. They describe themselves as a nonprofit organization responsible for “created and operated exclusively for religious, charitable, scientific, literary, or educational purposes and responsible for physically maintaining, repairing and securing the Hollywood Sign”.
https://www.10news.com/news/fact-or-fiction/fact-or-fiction-hollywood-sign-catches-fire-amid-la-blazes
https://www.latimes.com/california/story/2025-01-10/debunking-social-media-fire-myths-no-the-hollywood-sign-didnt-burn
https://www.cbc.ca/kidsnews/post/watch-fake-l.a-wildfires-posts-are-tricking-people-how-to-spot-whats-not-real
https://www.hollywoodsign.org/about-the-trust