Load the first file into a hash instead of reading the a file over and over again.

And what's with using cat?!?!?!

#!/usr/bin/perl

use strict;
use warnings;

my $File1 = '...';
my $File2 = '...';
my $File3 = '...';

my %keep;
{
   open(my $fh_keys, '<', $File1)
      or die("Can't open key file \"$File1\": $!\n);

   while (<$fh_keys>) {
      chomp;
      $keep{$_} = 1;
   }
}

{
   open(my $fh_in, '<', $File2)
      or die("Can't open input file \"$File2\": $!\n");

   open(my $fh_out, '>', $File3)
      or die("Can't create output file \"$File3\": $!\n");

   while (<$fh_in>) {
      my ($key) = /^[^|]*\|([^|]*)/;
      print $fh_out $_ if $keep{$key};
   }
}
[download]

Update: Added missing chomp as per reply.
Update: Added missing "$" in "my $fh_in" and "my $fh_out".