DR site SQL Server DB server not need Quorum Disk?
If Production site SQL DB Servers are down, will Windows Server Failover Cluster 
  be Online or Offline ?


If you have a setup that relies on a quorum disc, this must be available - ergo offline. This is a non recommended setup. Whoever set it up did not bother to validate it makes sense.

Geographically dispersed clusters are a proven technology (start at https://technet.microsoft.com/en-us/library/cc757840(v=ws.10).aspx) but their setup was seriously bothered here.

"The quorum data must be synchronously replicated between the sites. To ensure that the Server cluster guarantees of consistency are met, the cluster database must be kept consistent across all nodes. If the quorum disk is replicated across the sites, it MUST be replicated synchronously."

Simple like that.

If i were you, I would start retiring the error prone outdated approach and work towards AOAG with a read only replica offsite.

https://www.simple-talk.com/sql/database-administration/sql-server-2012-alwayson/

This eliminates the shared storage even locally (which IS a problem - shared storage means a dying server crashing the database structure takes down the cluster). The DR scenario is a complicated one, though. You likely end up having to have the quorum disc replicated.