How long will items stick around when dropped?
Based on past animal crossing games and judging from a week's worth of experience, items will persist forever when dropped until picked up, and they will be moved to the "lost and found" (does this exist in AC:NL? I haven't seen one yet) as soon as an event or new house needs to occupy the space the item took. In the original AC I left items all over several acres of land and they were only removed when (oops) an event cleared the acre.
I haven't noticed any dropped items disappearing so far. Note that buried fossils and gyroids are an exception to this rule; naturally placed buried items will be removed the next day.