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.
What you can do is that you follow Rashad, the man who is treating Duke Eltan when he leaves the Flaming Fist headquarter in the evening and see where that will end.
Or you can immediatly enter the headquarters and kill anyone between te party and Duke Eltan. When you reach the first 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 until he is recovered from the loyal followers of Sarevok.
Now it's a good time to open to lockpick all containers there and see what kind of valuables are hidden, which includes a Medium Shield +1 and a Splint Mail +1.
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.