Can a Crown Become Loose? If So, Would It Need to Be Removed and Re-cemented? Would I Need a Root Canal?

If you had work done on the tooth next to it and you experienced some discomfort after that you need to determine if it is the crown or the adjacent tooth that is giving you those symptoms, I would advice you to go back to the dentist that prepped the crown so he can assess the condition of it and determine which is the source of your discomfort, it is very uncommon for a crown to move, so before it becomes a bigger problem have it checked, most professionals would take into consideration that it is a recent restoration and will take that into consideration.

It sounds as if the symptoms you are having suggests that the nerve is going bad. In that case a root canal would be indicated.  It is possible that the crown has lost cement and is loose. This could cause sensitivity.  You should see your dentist again.  At this point I don't think he or she has done anything wrong.