Instead of having a hash for each directory, you could have a hash of hashes:

#!/opt/perl5/bin/perl -w
use strict;
use File::Slurp;

# The production files will be listed in
# the source hashes
my $report = "/home/mis/tstanley/SourceDiff.rpt";

my %input = (
  iqs_source => '/dsmpayroll/iqs-source',
  prg_source => '/dsmpayroll/prg-source',
  spg_source => '/dsmpayroll/spg-source',
  tpr_source => '/dsmpayroll/tpr-source',
  iqs        => '/dsmmigrate/development/iqs',
  prg        => '/dsmmigrate/development/prg',
  spg        => '/dsmmigrate/development/spg',
  tpr        => '/dsmmigrate/development/tpr',
);

my %data;

foreach my $key (keys(%input)) {
  my $dir = $input{$key};
  chdir($dir);
  foreach my $file (read_dir('.')) {
    my $sum = `sum $file`;
    my ($x) = split(/\s+/, $sum);
    $data{$key}{$file} = $x;
  }
}

## Open the report file
...
[download]

Update: It seems there's no need to build a hash of hashes, since we only need to look at a pair of directories at a time. I elminated the pesky chdir commands while I was at it. Here it is a version with (just about) all redundancy eliminated, but without any change in functionality:

#!/opt/perl5/bin/perl -w

use strict;

# The production files will be listed in
# the source hashes
my $report = "/home/mis/tstanley/SourceDiff.rpt";

sub DEV  () { 0 }
sub PROD () { 1 }

my %input = (
   IQS => [ '/dsmmigrate/development/iqs', '/dsmpayroll/iqs-source' ],
   PRG => [ '/dsmmigrate/development/prg', '/dsmpayroll/prg-source' ],
   SPG => [ '/dsmmigrate/development/spg', '/dsmpayroll/spg-source' ],
   TPR => [ '/dsmmigrate/development/tpr', '/dsmpayroll/tpr-source' ],
);


## Open the report file
open RPT, ">$report"
or die "Unable to open $report: $!\n";

my $date=`date`;
print RPT "$date\n\n";


## For each set of directories, we find the common file names in them,
## and then go through those files and compare the sums. If the
## Production file is different, we write it out to the report file.

my $print_seperator;  # False for first loop iteration.

foreach my $key (sort keys %input) {
   local *DH;
   my $dir;
   my $file;
   my %dev_crcs;
   my %prod_crcs;

   $dir = $input{$key}[DEV];
   opendir(DH, $dir)
   or die "Unable to open dir $dir: $!\n";
   while (defined($file = readline(DIR))) {
      next if $_ eq '.';
      next if $_ eq '..';
      next if $_ eq 'cob';
      next if substr($_, -4) eq '.ffl';
      next if substr($_, -4) eq '.lst';
      $dev_crcs{$file} = 0+`sum $dir/$file`;
   }

   $dir = $input{$key}[PROD];
   opendir(DH, $dir)
   or die "Unable to open dir $dir: $!\n";
   while (defined($file = readline(DIR))) {
      next if $_ eq '.';
      next if $_ eq '..';
      next if substr($_, -4) eq '.ffl';
      next if substr($_, -4) eq '.lst';
      $prod_crcs{$file} = 0+`sum $dir/$file`;
   }

   if ($print_seperator) {
      print RPT "\n\n";
   } else {
      # True for all but first loop iteration.
      $print_seperator = 1;
   }

   print RPT "$key Source Files\n";
   print RPT ("=" x length($key)) . "=============\n";

   foreach $file (keys(%dev_crcs)) {
      if (exists($prod_crcs{$file})
      && $prod_crcs{$file} != $dev_crcs{$file}) {
         print RPT "$dir/$file\n"
      }
   }
}
[download]

Keep in mind that files in one directory but not in the other are not printed, just like in your solution.

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