You haven written curly brackets instead of round brackets to capture the match into "$1".

For excluding "vfiler0" block you'll have to identify where new block begins. It may be a blank line.

...
my $vfiler0 = 0;
while(<FaH>)
{
  chomp;
  if(/^\s*$/)
  {
    $vfiler = 0;
  }
  if(/vfiler0/)
  {
     $vifiler = 1;
  }
  if (!$vfiler && m/([0-9]{1,3}\.[0-9]{1,3}\.[0-9]{1,3}\.[0-9]{1,3})/)
  {
    $ip=$1;
    print FaHout "$ip\n";
  }
}
...
[download]

You want to use round brackets to capture to $1, not curly brackets. Be sure to copy and paste code instead of retyping in order to avoid typos.

You also need to tell your script what you mean by vfiler0 removal. It sounds like you meant to parse the file as records, not line by line.

Think about how you decide what to keep... in slow motion.

1. How far do you get before you're sure you want to ignore stuff?
• What made you decide to start ignoring things? Write a regex to match that.
2. Where do you go next from there?
• What rule do you use to stop ignoring stuff?
• There should be a unique way to identify when a new record is starting.
• Write a regex or string comparison to match that.
• If it crosses multiple lines, you will want a variable to track what state your match has reached so far.
3. Make an ignore/keep flag to set or clear when you decide to process or ignore the lines as above.
4. For each new line, check the flag before looking for IP addresses to print.

Learning to simplify and break down your own thought processes into an algorithm that the computer can follow is a very important skill.

You'd be much more likely to read the file reliably if you take advantage of the format and keywords.

If you look for the line storing IP addresses (the ones beginning with "IP address" you are guarenteed to get all of the IP addresses. On the other hand, if you scan for what matches an IP4 syntax, you have two risks. (1) you are just crossing your fingers that only lines that actually have IP addresses store values that look like IP addresses. (2) I'd also note that your IP address pattern assume IP4, but can you really be sure you won't have a few entries in IP6 format?

I'm guessing from your sample that your format rules look something like this:

• Records are delimited by blank (all whitespace) lines
• The first run of non-whitespace on the first line of a record is its id
• The remaining lines consist of attribute value pairs: attributeName: value

To pick out IP addresses based on attribute names, you would use a very simple state machine that determines the current record based on the value of $sId and the current attribute based on the value of $sAttribute, like this:

use strict;
use warnings;

my $sId='';
my $sAttribute='';
my $sValue='';

while (my $sLine = <DATA>) {
  chomp $sLine;

  #figure out the id of the current record
  if ($sLine !~ /^\s*$/) {
     # $sId is '' when we read the first line
    if (!$sId) {
      # first line of record contains id, e.g. vfiler0, vfiler1, etc
      # id is first run of non-white characters - store it in $sId
      ($sId) = ($sLine =~ /^\s*(\S*)/);
    }

    #skip all vfiler0 record lines
    next if ($sId eq "vfiler0");

    #get IP addreses for other records if attribute is "IP Address"
    #attribute name goes from first non-white character to first ':'
    ($sAttribute, $sValue) = ($sLine =~ /^\s*(\S[^:]*):\s*(\S*)/);

    if ($sAttribute && ($sAttribute eq 'IP address') && $sValue) {
      print "$sValue\n";
    }

  } elsif ($sLine =~ /^\s*$/) {
    # records divided by entirely blank lines
    # when not in record set id to ''
    $sId='';
  }
  #print STDERR "line ($sId): <$sLine>\n";
}
[download]

open (FaH, "/tmp/vflist1.out");
my $flag = 0;
while ( <FaH> ) {
chomp;

if($flag == 0)
{
      if($_ =~ /vfiler1/)
      {
          $flag = 1;
      }

      if($flag == 0)
      {
          next;
      }
}

if (m/{[0-9]{1,3}\.[0-9]{1,3}\.[0-9]{1,3}\.[0-9]{1,3}}/) {
                                $ip=$1;
                                print FaHout "$ip\n";
               }
}
[download]

#!/usr/bin/perl
use strict;
use warnings;
use Regexp::Common qw /net/;


$/ = "";

while (<DATA>){
        next if /\A vfiler0 /x;
        /$RE{net}{IPv4}{-keep}/ and print $1, "\n";
}

__DATA__
vfiler0                          running
    ipspace: default-ipspace
    IP address: 10.42.156.16 [e0a]
    IP address: 10.42.156.19 [bae_test]
    IP address: 10.42.156.18 [e5a]
    Path: / [/etc]
    UUID: 00000000-0000-0000-0000-000000000000

 vfiler1                          running
    ipspace: default-ipspace
    IP address: 123.123.123.123 [unconfigured]
    Path: /vol/palermo [/etc]
    UUID: 4fa0d3d8-ecd6-11df-af94-00a09808463c

 vfilerbae                        running
    ipspace: default-ipspace
    IP address: 123.124.123.123 [unconfigured]
    Path: /vol/bae_testing [/etc]
    UUID: 605f7aa8-ecd6-11df-af94-00a09808463c
[download]

Output:
123.123.123.123
123.124.123.123

hth,
PooLpi

        open (OUTDATA, ">> $vfile2"); 
        open (GETDATA, "$vfile"); 
        my $vfiler0 = 0;
        my $vfiler = 0;
        while(<GETDATA>)
        {
        chomp;
        if(/^\s*$/)
            {
            $vfiler = 0;
            }
        if(/vfiler0/)
            {
            $vfiler = 1;
            }
        if (!$vfiler && m/([0-9]{1,3}\.[0-9]{1,3}\.[0-9]{1,3}\.[0-9]{1
+,3})/)
            {
            my $ip=$1;
            print OUTDATA "$ip\n";
            }
        }
[download]

10.42.156.16
10.42.156.19
10.42.156.18
123.123.123.123
123.124.123.123
123.123.123.123
123.124.123.123
[download]

small syntax error, got it to work. Thanks Much!

while(<>) {
   next until /vfiler1/;
   print (split)[2], "\n" if /^IP address:/;
}
[download]

which would consume lines until the "vfiler1" line, then split lines beginnning with "IP address" on whitespace and print the third field - the address you are looking for.

Not tested! :)