How can a CFI renew their certificate with the FAA during the government shutdown?
Short answer is no, but there is a way to renew your CFI while Federal Government is shutdown. However, it's not through the FAA because so long as the FAA is shut down it cannot process applications.

(After a few hours of freaking out at the thought of having to redo my CFII check ride, I discovered there is a solution to this. 

While the FAA is not processing applications during the government shutdown, the FAA has authorized certain non-federal employees to accept and process airmen applications for CFI renewal AND to issue temporary flight instructor certificates. These are called Airman Certification Representatives (ACR), and are defined as:


  Airman Certification Representative (ACR). An ACR is an individual representing a specific FIRC provider who is authorized under Title 49
  of the United States Code (49 U.S.C.) § 44702(d) to accept airmen
  applications for renewal of a valid flight instructor certificate from
  successful graduates of that provider’s FIRC and to issue temporary
  flight instructor certificates


Source: Advisory Circular 61-83J.