Q:

How to convert local date-time string to Unix timestamp (GMT)? 

time_var="6/23/2016 3:20:00 AM"

(this is in EDT)
We need to get unix timestamp for this variable after converting its value to GMT. 

A:

Just use the -u flag while passing the date with -d:
$ time_var="6/23/2016 3:20:00 AM"
$ date -d"$time_var EDT" -u
Thu Jun 23 07:20:00 UTC 2016

Note I also appended EDT to your date. 
From man date:
   -d, --date=STRING
          display time described by STRING, not 'now'
   -u, --utc, --universal
          print or set Coordinated Universal Time

