What would happen if I don't get a crown on my root canal after 6-8 weeks?

The biggest problem with delay is the risk of fracture or loss of seal.  If the temporary is lost you could have leakage that can cause it to fail or need re-treatment.

Hi:As long as the tooth has not cracked, then ther should not be any problem getting the permenant crown after 8 weeks.Best,Dr. Maddahi

The
temporary filling material in your tooth is intended for short term use only,
in my opinion it’s already been too long. You don’t need to have the crown done
tomorrow, but you should have a permanent filling put in that tooth as soon as
possible. Waiting until October is just asking for a problem. If that tooth
breaks, at a minimum you’ll need a larger more expensive filling than you do
right now.

If the tooth is asymptomatic I would think about having it restored right away.  The longer you wait the more at risk the tooth may fracture and require removal.  Typically the tooth after a root canal can become more brittle and fracture easier than before, this is encouraged when there is lot of missing tooth due to the large decay that was removed.  The temporary filling provides basically zero support to the surrounding tooth. If it is a financial reason you are putting off the crown, I would highly encourage you to at least have the temp filling removed and replaced with a permanent buildup/filling.  The buildup will at least provide a structural support and foundation to the remaining tooth structure.  In addition, if the dentist can remove the bite off your tooth a little ways then it will take more force off of the tooth and lessen the risk of fracture (this is no guarantee though). The permanent crown, however, will give you the best support and prevention of tooth fracture.   

If the root canal is finished, there won't be pain because there is no nerve to feel it.  The risk you have is fracture of the tooth because it has been greatly weakened by the decay and the removal of tooth necessary to do the root canal procedure.  Best to get in ASAP