Let's say file.txt contains the following text.

removeMe
Hello
World
removeMe
Hello
World
removeMe
Hello
World
removeMe
Hello
World

Remove line 1

In this example, the entire file will be print, except for the very first line. next if 1 == $. means to that line 1 is the current line, and to then go onto the next line.

$file = "/path/to/file.txt";
open(FH, '<', $file) or die "cannot open $file";

while (<FH>) {
  if (1 == $.){
    next
  };
  print $_;
}

close(FH);

Running this script will produce the following output.

Hello
World
removeMe
Hello
World
removeMe
Hello
World
removeMe
Hello
World

Remove every third line

With the following Perl script removes line 1 in the file, and then removes every third line thereafter.

$file = "/path/to/file.txt";
open(FH, '<', $file) or die "cannot open $file";

while (<FH>) {
  if (1 == $. % 3){
    next
  };
  print $_;
}

close(FH);

Running this script will produce the following output.

Hello
World
Hello
World
Hello
World
Hello
World

Add a new line every fourth line

$file = "/path/to/file.txt";
open(FH, '<', $file) or die "cannot open $file";

while (<FH>) {
  if (1 == $. % 4){
    print "\n"
  };
  print $_;
}

close(FH);

The result.

removeMe
Hello
World
removeMe

Hello
World
removeMe
Hello

World
removeMe
Hello
World