My crown keeps falling off a tooth I had a root canal done on 8 months ago. Should I keep the tooth or extract it?

If the xray reveals a well done root canal, then consider that the crown is the problem.  Either a core build up around the post or a new post may be needed for better retention.  Seek another opinion but bring the xrays.

This question is tough to answer without an x-ray, however if the crown has come off 3 times in 8 months it sounds as if there is a problem with the amount of tooth structure remaining. If you decide to have the tooth pulled you need to have a plan for how you are going to replace this tooth. Are you going to have an implant placed or can a fixed bridge be placed instead? Talk to your dentist about this before making the decision to pull the tooth. Find out what all your options are and also the cost involved for each one. Both you and the dentist are going to get frustrated by repeated return visits to recement this crown.

It is very hard to say what you should do with the tooth without more information like an x-ray.  The answer to your question depends on many factors.  It sounds like you trust your dentist.  I would get a consult with them and see what your options are.  Good luck!