(I already put Spoiler in the title so I won't bother adding spoiler tags)

I have just finished the quest for Prince Lhotun in Sentinel where I was supposed to find out what happened to his brother. Went to the dungeon, found the death certificate, and showed it to the Prince.

He acknowledged my efforts, gave me my reward, yet the entry in the quest log still remains. The last entry related to this quest is the one that goes something like "I met the secret agent of the Underking" etc etc (it's the entry that reveals the name of the dungeon I have to travel to for the death cert).

Is this a known problem, or is this one of those infamous DF bugs? Is there anything I can do to remove this quest log entry?

It may be normal. Some quests have a period before being declared closed. A kind of cooldown for avoiding side effects like the immediate disappearance of an NPC under the very eyes of the player !
For example, in the first quest (Lady Brisienna), the quest is technicaly closed one day after the meeting. If this was not done so, Brisienna would disappear immediately after the dialogue (*pouf*)
Usually the delay is 1 hour or 1 day... An other side effect is to permit a rereading of the log...

I do not know if this is understandable, sorry for my english!