Yes, it does get left behind.
Descending rings are meant to be used to facilitate the recovery of ropes, they save your rope from getting horribly dirty, damaged, or stuck, and leave much less of an impact than rappelling off of a tree or branch.
Pulling you rope off of a tree will saw into the trunk and leave permanent scars, it could even possibly lead to the eventual death of the tree.
Yes it leaves a trace, but they're fairly inert articles, and it's unlikely that anyone will ever find your sling except for fellow alpinists, they're only typically used for fast descents off long alpine routes in places where there's no other way down (aside from down-climbing, which is how most climbers get into accidents).
If an alpinist finds a descending ring while on a descent, the proper ediquette is to cut the webbing, retire the ring and replace it with one of your own–as it's never advisable to use climbing gear that you don't know the history of–but usually they will inspect the ring and the webbing to see if it's still usable. Someone my reuse it if they feel that it is safe, sometimes they will reuse the ring only but add a sling of they're own, leaving the original sling as a backup. This is when things get messy, when old slings start to pile up, but some responsible climber eventually comes along and cleans the mess.
When rappel anchors pile up: