Is there a ruling that states this is the case? It seems logical that you wouldn’t be able to, but I can’t find anything in the code that specifically states that you can’t.
Shipwrights only change where you can take a repair action, but they don’t provide any mechanism for actually removing the fire mast first.
You can use one to repair other masts on the ship, but if you try to repair a mast that’s on fire it’ll remain on fire.