I got rid of the cake, it's still happening.
I click on the damaged items and select "replace", it's still happening.
I have been using the sim minute or three between fires to move as much of the furniture to the family inventory as possible; I don't want to lose upgraded items etc.
ETA: This started with a fire on another birthday cake, normal, which I deleted and then bought another cake, and then this started happening.
ETA again: Okay, I think I've found the problem. There's a square of floor tile that I cannot delete between fires or anything, and now that I've removed the rest of the floor tiles in the room, that's where all the fires are starting. I cannot get it to stop burning and I cannot delete the tile at all.
Also, at this point the firefighter has stopped fighting the fires. He's just standing there. My sims (and one guest who was over) are fighting all the fires now.
I have an idea: may I use edit town to build myself a new lot? Then I could just move the whole family and get away from this floor tile that won't stop burning.
ETA x3: may I try using moveobjects to delete the problematic floor tile?
I have tried to get the sims further away from the fire, but when they're on the way, or hanging across the street, they get the Fire! action again and I can't seem to get them far enough away to not come back (except for the two who are in school).