First, you may want to switch from calling chop() to chomp() instead (the latter will only strip off an end-of-line character, as defined by the variable $/).

Next you probably don't need the unnecessary variable $row, since both chomp() and split() operate on $_ implicitly.

Now, to answer your question, you can skip every other line by testing $. for odd/even-ness (use experimentation to decide which one -- odd or even -- you need to skip). For example, if the first line and every *other* line thereafter should be skipped:

open (DB, "<$data_file") or die "$data_file: $!";
while (<DB>) {
   next if $. % 2;   # change 'if' to 'unless' to skip even lines
   chomp;
   my @fields = split /\|/;
   print "$fields[0]";
}
close DB;
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dmm

If so, then I'll second dmmiller2k's answer.

If not, then you might want to consider using a regex to match the lines you want to omit. It will probably take more time than testing $., so you might want to use Benchmark, if that's a concern for your application. Assuming you want to omit the referer line that lists your own URL, you could do something like this:

open (DB, "<$data_file");                    
while (<DB>)                                        
  {
   next if m!www.yourserver.com/your/referer/here!;
   $row = $_;                               
   chop $row;                                 
   @fields = split (/\|/, $row);
   print "$fields[0]";
  }
close (DB);
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while(<DB>) {
  print ( split(/\|/) )[0];
}
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