This (untested) approach uses a cache to get the Sum of Squared Deviations to the mean one time. Your approach does this calculation for each call to the correlation sub. So this saves about 11 billion or more (more I think :-) )calculations.

#!/tools/bin/perl
use strict;
use warnings;
use List::Util 'sum';

my @keys;
my $probes;

open my $FILE, "data.txt" or die "ERROR: cannot read file $!\n";
while (<$FILE>){
    chomp;
    my ($key, @line) = split /\t/;
    $probes->{$key}{vals} = \@line;
    $probes->{$key}{mean} = sum(@line) / @line;
    push @keys, $key;
}
close($FILE) or die $!;

my $cache;
## Sum of Squared Deviations to the mean
for my $key (@keys) {
    my $mean = $probes->{$key}{mean};
    my $ss;
    for my $dat (@{ $probes->{$key}{vals} }) {
        $ss += ($dat - $mean) ** 2;
    }
    $cache->{$key} = $ss;
}

## Correlation Calculation
my $count = 1;
for my $i (0 .. $#keys){
    for my $j ($i+1 .. $#keys){
        my $cor = correlation($probes, $cache, @keys[$i, $j]);
#        $calProbes{$probesArray[$i]."-".$probesArray[$j]} = $cor;
        print $count++,"\t", "$keys[$i]-$keys[$j]\t$cor\n";
    }
}

## Sum of Squared Deviations to the mean
sub correlation {
    my ($probes, $cache, $key1, $key2) = @_;
    my $arr1 = $probes->{$key1}{vals};
    my $arr2 = $probes->{$key2}{vals};
    my $mean_x = $probes->{$key1}{mean};
    my $mean_y = $probes->{$key2}{mean};
    
    my ($ssxx, $ssxy, $ssyy) = ($cache->{$key1}, 0, $cache->{$key2});
    
    for(my $i = 0; $i < @$arr1; $i++){
         $ssxy += ($arr1->[$i] - $mean_x)*($arr2->[$i] - $mean_y) ;
    }
    
    return $ssxy/sqrt( $ssxx * $ssyy );
}
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Update: as moritz suggested, PDL::Stats::Basic might provide the optimal solution. My response is using plain perl.