Extreme sensitivity after temporary crown. Do I need a root canal?

You could have an irreversible pulpitis which would indicate a root canal is needed.  Have your dentist evaluate again to determine if you need this.1. Is your tooth sensitive to hot liquids?2. Does it linger or go away immediately?3.  Are you consistently taking Advil or Tylenol for the discomfort?  The answers to these questions are what your dentist will be looking for when determining whether or not your tooth will need a root canal.  

I am sorry that you are uncomfortable.  

If the pain is there all the time, it may be from the temporary not fitting well to or inflammation in the tooth.  If there was sufficient local anesthetic when preparing the tooth and it still hurt, that may also indicate inflammation in the tooth.  Root canal would take away the pain but there is a possibility that a well fitting crown may do the same.  Without exam and x-rays, it is difficult to tell.  

It may be advantageous to get a second opinion from an Endodontist (Root Canal Specialist)

Good luck