Try this, but beware of:

• lines that contain both the start pattern and the end pattern
• lines that contain characters before the start pattern, which are not handled here, as aren't lines that contain characters after the end pattern
• start and end patterns that may span a line
• errors opening files - you might want to check for them
• the \n in the print command is redundant, as the original \n was not chomped off

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

my $start_pattern = '<!-- start section-->';
my $end_pattern   = '<!-- end section -->';
my @files_to_look_in = ('/path/to/files1.html', '/path/to/files2.html'
+);

my $write_line = 0;
foreach (@files_to_look_in)
{
  open HTM_FILE, "<$_";
  while (<HTM_FILE>)
  {
    $write_line = 1 if /$start_pattern/i;
    $write_line = 0 if /$end_pattern/i;
    print $_, "\n" if $write_line;
  }
  close HTM_FILE;
}
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