This game has very little randomness. Play the game repeatedly to learn what happens at each moment. Try doing things a bit differently at each event to see what causes what. Anticipating the game is insanely helpful for speedrunning it.

Play close attention to each type of spray: stream nozzle, fog nozzle, (extinguisher), firetruck. Notice for each spray how wide of an area fires can be put out, how long it takes to put them out, and which distances they work best. Generally, it's fastest to run up close to many fires and spray the fog nozzle to put them all out. It helps to circle around the area a bit to prevent stragglers. Again, learn how long it takes to put out fire so you don't move on too quickly.

Only extinguish fires when necessary! If you need to cool yourself down, or a character can't be pushed through the fire (or is too slow when being pushed), or the level explicitly requires it, put them out. Otherwise, it's usually faster just to go straight through and ignore it.

Find a good balance between speed and thoroughness when extinguishing fire. Slower means less likely to miss any stray fires, but at the cost of time. Different situations need different levels of thoroughness. If the fire's out in the open, it'll be harder to miss, so go a little faster. If you're afraid you'll miss one, take your time. Waiting for a small fire to become big enough to see kills your time. Again, be aware of which spray is best in each situation!

Anticipate things that need to be chopped. When walking towards collapsed beams, locked wooden doors, etc., time your first swing so it lands right when you reach the object.

There are many times where you have to wait for something. There's usually stuff you can do while waiting, which can save time. For example, if you're waiting on dialogue, why not get FCDs or put out fires? Get efficient!