Save Duke Eltan is a main quest, it can be started in chapter seven after hearing from many different sources which all say Duke Eltan has been ill before Scar's assassination and they all suspect foul play, but they don't have any evidence for it. The sources from which you can hear this are: Tamoko, Delthyr and many more, however, despite how Delthyr tells you that you should follow the suspicious healer Rashad who leaves Flaming Fist headquarter every night, you can't do that as there're no scripts allowing him to spawn outside the building.
Head directly into the Flaming Fist headquarters and kill anyone between the party and Duke Eltan. Alternatively, sneak in with a strong thief or someone strong and invisible. If you are hiding or invisible, you can avoid a fight with the enemy upstairs too. When you reach the second floor where Duke Eltan normally is, you find him and the man treating him, Radesh, who after a conversation turns out to be a greater doppelganger trying to kill the party. After he is killed, Duke Eltan will ask of you to bring him to the Harbormaster, who is good friend of Eltan and can hide him from the loyal followers of Sarevok until he is recovered.
When Eltan is brought to the Harbormaster, the Harbormaster will hide him and promises to take care of him. The party gains as a reward 2500 experience points and has probably done something good for the future of the city.