Try it this way. Provided I haven't made any elementary mistakes, it shoudl produce the same result and run in a fraction of the time:

#!/usr/bin/perl -w
use strict;
use warnings;

if( @ARGV < 3){
      print "usage: A message here\n";
      exit 0;
}

open(INPUT1,$ARGV[0]) || die "Cannot open file \"$ARGV[0]\""; #Orginal
+ IDs and four letter codes
open(INPUT2,$ARGV[1]) || die "Cannot open file \"$ARGV[1]\""; #Orginal
+ IDs in the second column
open(RESULTS,">$ARGV[2]")|| die "Cannot open the Results file \"$ARGV[
+2]\""; # Origanl IDs will change to four letter code

my %origins;

while (<INPUT1>) {
    chomp;
    my @columns = split '\t';
    $origins{ $columns[0] } = $columns[1];

}

close(INPUT1);

while( <INPUT2> ) {
    chomp;
    my( $bioC, $contig_id, $pip ) = split("\t", $_);
    print RESULTS "$bioC\t$origins{ $contig_id }\t$pip\n";
    }
}
close(INPUT2);
close(RESULTS);
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How small a fraction? If input file 1 contains 1000 lines, then < 1/1000th of the time