For the elves who lived in the woods, there would have been little or no problems with sanitation. There still are tribes that live in the rainforest today and activities like defecation and urination can be carried out in a forest leaving little or no trace.
With regards to the cities, it would be wrong to extrapolate from what we know about medieval Europe to a fantasy/medieval setting. In the medieval Islamic world, cities were supposed to have had "intricate systems of sanitation, public baths and fresh-water supplies"
http://www.laphamsquarterly.org/essa...e.php?page=all
I suppose we could assume that cities like Minas Tirith may have had similar systems of sanitation? Of course they would not have been as clean as our own modern cities, but maybe not as bad as the European cities that were devastated by the plague...