I have just experienced the following bug:

During my travels i have picked up a number of bard instruments, including /rjorn's drum and Finn's Lute. I brought these items with me because i stimbled upon them, and did not have the quest. When i got the quests at the bard college and handed them in, i got rewarded fine.
However the goals "Find Rjorn's drum" and "Find Finn's lute" were not removed from my journal.
Expected: The goals should have been removed from my journal, as i found them before getting the quest.

If you like having as clean a quest log as possible with Skyrim, I've found it is better to NOT pick up quest items unless you have a quest for it for the problem you state. Yes, that means backtracking to places you've been - but that's the trade off, a clean journal with backtracking or a messy journal without backtracking.