my testdata was like this:

snp.txt:
-----------
rs34569384
rs123456
rs234567
rs753444
----------

chr22.txt:
---------
bla sijghs bla
rs234567 yes,first one
fdjg
rs123456 yes, second one
---------
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A part for the lenght of the row in file 2 , I do not see any difference. I guess you used tab to split the rs234567 with the others, but since we are looking at the beginning I do not think it makes a big deal. On the other hand my snp.txt doesn't look like that, but it is a collection of rs separated by a space with print "@output" or just a sequence of rs if print @output. So my file1 looks like:
rs234567rs265897rs2458796rs2658974rs...

Should I split the rs with /n then print a file 1 and then start from there? In this case I will have a file that looks like yours but that will be a collection of line, no an array

??? Your first paragraph makes no sense. You can't tell what snp.txt looks like if you print @output. The contents of @output is processed, changed by a split and a grep. Use an editor or 'less' to look at a file.

You might try out the following (this is the same program, just added a Dumper-line before the foreach:

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

use Data::Dumper;

use strict;
my$line;
my@fields;
my@output;
open (FILE1, 'snp.txt') or die "can't open the file: $!";
open (FILE2,'chr22.txt') or die "can't open the file: $!";
open (FD, '>test.txt') or die "can't open the file: $!";
my $position = tell(FILE2);
my %rs;
while ($line=<FILE2>) {
  my ($key)= $line=~/^(rs\d{5,})\b/;
  if (defined $key) {
  $rs{$key}= $position;
  }
  $position=tell(FILE2);
}

while (defined ($line= <FILE1>)) {
    my@fields= split (/\s+/ ,$line);
    push @output, grep /^rs\d{5,}\b/ ,@fields;
}

print Dumper(\@output,\%rs);

        foreach (@output) {
              if (exists $rs{$_}) {
                seek(FILE2,$rs{$_},0);
                my $line= <FILE2>;
    print FD $line;

    }
}
close FILE1;
close FILE2;
close FD;
[download]

With the data I posted, I get the following output:

$VAR1 = [
          'rs34569384',
          'rs123456',
          'rs234567',
          'rs753444'
        ];
$VAR2 = {
          'rs123456' => 43,
          'rs234567' => 15
        };
[download]

As you can see @output (==$VAR1) contains a lot of rs-numbers. %rs (==$VAR2) contains some of the same rs-numbers and corresponding file positions. You should see the same if you use my data.

Now try it with your data. What is different? Are there rs-numbers that are in both @output and %rs? Do the file position numbers look correct?