Since I've started playing since the changes to fire, I have seen this problem again and again.
To engineers and excitable pilots, do not panic when you see that something is on fire. Most notably the hull component. Just because you see a fire, that does not mean you should automatically pull out your extinguisher and put it out or complain at the engineer.
As you should know, extinguishers and repair tools share a cooldown. The cooldown the extinguisher inflicts is three seconds, which is three seconds that you can not repair the ship's armor. Before you use that extinguisher, you need to ask yourself a question. "Is it likely that the armor will be brought down in the next three seconds?" If there is a good chance that it could be, such as if you are at one red bar and you have one stack of fire, then you are better off repairing that armor instead. Failure to do so can result in the death of your ship and possibly the ire of the engineer who was waiting patiently right there next to the hull to properly repair it.
Once you have a safe margin of armor, you are free to extinguish the fire. However, there are a couple more variables to consider. At a certain number of stacks of fire, it becomes nearly or completely impossible to repair through the increasing damage. For example, if you have 20 stacks of fire, you should probably extinguish it no matter how low the armor is and just hope you take no more incoming damage. You should also consider how much incoming damage you're taking before you consider your margin of armor safe, especially with powerful howitzer shots and withering gatling fire.
For shrewd pilots looking to take advantage of this widespread problem, I would recommend considering the Hades for its unique properties.