The objective of Tre Kronor Infernum is to score victory points and there are several ways to do so.
Connect servants to rescued room
Save possessions from rescued rooms
Steal possessions from burnt down rooms
The challenge here is that you do not know in advance which rooms will be resuced and which will burn down so when the heat literally goes up, Tre Kronor Infernum will turn from a strategical game to a tactical. Nevertheless, the game can be divided into three specific phases.
As the game starts, there is only one fire in the castle and the fate of each room still uncertain. The objective of this phase is to place your servants at optimal spots. What is an optimal spot then? The answer is a spot where one servant reach as many rooms as possible. Since one servant is enough, more servants in the chain is a waste of time and resources. In the image above, the single red servant in the bottom left chain is connected to four rooms; A1, A2, B1 and B2.
This may create "negative races" where players avoid placing a second unnecessary servant in a chain. The negative race is particularly common in 3 player games, since 4 servants are required to connect to the rooms closest to the wells. It is important to stay calm and find alternative spots and let someone else waste a second servant in the chain. However, there are two cases where it may be beneficial to place a second servant.
You place a second servant to deny another player a place in the chain.
You place a second servant to cover a servant in an entrance of a burning room (which may have to be removed if the fire spreads outside of the room)
In the previous image, the green player missed out the chain in the upper left corner and will not score for the room, nor be able to save/steal from it. In the lower left corner, there are two green servants outside entrances so even if one is removed due to a fire, there will still be a green servant in the chain (provided that the broken link is restored).
With all servants strategically placed, it is time to choose from which rooms to save and from which rooms to steal. You have a limited number of markers so don't rush the decision. Instead, observe how the other players play their save/steal markers and fire spread markers to get an idea of their intentions. Generally, a room with only one fire is a candidate for saving and a room with two fire/ash a candidate for stealing. However, if other players specifically target that line or row with their fire markers, saving may be a bad idea and vice versa. If possible, try to avoid saving and stealing from rooms along the same column or row. This is because you have to play fire markers to determine the column or row that a fire spreads to and it will be more difficult to target your fire against "steal" rooms if you mix "save" and "steal" rooms in the same column or row.
Extinguishing fires is another possible action. Use the action to protect your interests by rescuing the rooms you want to be saved and leaving the rooms you don't want to be rescued. However, do this with caution; you don't want the other players to find out your interests. Otherwise, you will have to do all the extinguishing yourself and defend against targeted fire markers. Likewise, you should try to find out the other players' interests. If another player wants to rescue "your" room, let her waste actions on it. If other players are eager to rescue a room, a targeted fire will damage their score greatly.
With most of the save/steal markers on the board, it's just a matter of closing the game by extinguishing the last fires. Don't do the mistake to play this phase with casual moves. With a well played (i.e. observed) mid game, you should have an idea of who's in the lead and take your last chances to dethronize them. Are there still rooms that you are not interested in rescuing? Remove your servants from the chains connecting them. It's also a good idea to keep one or possibly two save/steal marker to the endgame, since it should now be easier to predict which ones will be rescued and which one will burn down. However, don't keep too many as you may run out of time or good rooms to place them in.
Tre Kronor Infernum is not so much about your game as it is about the other players' games. If you know how they play, you will know how to play yourself.