Well, the short answer is, sometimes they probably did. Some whaling vessels were "lost at sea" and never heard of again; we don't know if they were sunk by storms, stove in by angry whales, kidnapped by aliens, experienced a fatal epidemic, or suffered a disastrous fire.

But all sailors and shipbuilders were well aware of the dangers of fire onboard a wooden ship, and so the ships were designed to minimize the risk. If you were trying out (rendering) the blubber on shore, you could just use an iron try pot over any old fire. Whaling ships, however, went to sea with a brick furnace on the deck; the fire was built inside the furnace, and heated the pots on top. (The setup is called a "tryworks".) So the fire was contained within the brick furnace. The bottom of the furnace was made of multiple layers of brick, and also air gaps, which are insulating, so that where the furnace sat on top of the wooden deck, the bricks wouldn't get hot enough to ignite the wood.

As a bonus, cooking fires could be dangerous because there was probably only one person working in the galley, who might be inattentive. But the tryworks were right out on the deck in the middle of the whole crew, who were busying doing the trying-out, so there was little chance of a small fire forming and going unnoticed until it had become a large fire.