Spent about 4 years on a LA class sub (SSN-700)
What can burn - There is a lot things that in a sub that can burn under the right conditions.
There is a large diesel engine up front with oil and its day tank.
The interior of the hull is insulated and if heated hot enough can start decomposing into some bad stuff.
Bedding, plastics, title floors, electronics...
Its in a ship yard for overhaul and the hull status is not indicated here. The hull might have huge sections removed to allow access for removal, installing or replacing large pieces of equipment. Even if the hull was intact, closing the hatches might not work, inside there are oxygen and high pressure air systems that if effected would feed the fire.
Most likely what happened was a big fire started in an area and got everything really heated up. Fire moved into the lagging that is wrapped around everything to contain heat and noise and moved into hard to reach areas. The big fire that started the whole mess could have been taken down in a reasonable time but the burning lagging took a extended time to clear.
Fire suppression systems? Small fire extinguishers to handle a fire and if the fire is to big for it you move to the 3" fire hose that are installed through the boat. Its not like a controlled room where you can pull the halon switch; You got people sleep, eating, living inside the equipment room.
A sub is a interesting beast, there were so many ways that if a situation was not handled right you would end up dead. Remember several discussion about what would be the quickest way that you could die while at sea - we figured that a rocket ignition or a hull collapse at TD would be the best ones in that your nerve signals would not reach your brain before you were dead (under 0.4 seconds).