From what I can tell, you are checking everything after the first field matches. The best way to do this is to put the rows of the first file into a hash and then use the rows of the second file to do a lookup to see if the hash entry exists.

The following code is not guaranteed to run (I had a long night last night!) but should show the general idea....

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

open (INA, $ARGV[0]) || die "cannot to open gene file";
open (INB, $ARGV[1]) || die "cannot to open coding_annotated.var files
+";

my @sample1 = <INA>;
my @sample2 = <INB>;

# use map for this maybe?
foreach my $line (@sample1) {
    my ($id, $rest) = split( '\t', $line, 2);
    chomp ($rest);
    $hash1{$rest} = $id;
}

foreach my $line (@sample2) {
    my ($id, $rest) = split( '\t', $line, 2);
    chomp( $rest);
    if (exists($hash1{$rest}) {
       print "Match: $line\n";
    }
}
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