Do I stay on fire if I save and reload the world?

In short? Yes.

With latest updates (specifically Minecraft 1.14) they’ve managed to fix this. Meaning there’s no point in trying to escape fire with reload commands or even by restarting server.

However in old versions of Minecraft this would still work, so in some cases you could simply /reload the server and poof the fire effect is gone.

Of course reloading the world would bring other issues like lag which would be noticeable by other players.