open(FILE1, $ARGV[0]) || die "Error: $!\n";
my $save=0; 
my $saveto=''; 
while (my $line=<FILE1>){
  chomp $line; 
  my @parts=split(' ',$line,2); 
  if ( (!$save) && $parts[0] eq 'X1') { $saveto=$parts[1]; $save=1; } 
  elsif ($parts[0] eq 'X2' 
      ||($parts[0] eq 'BEGIN' && $parts[2] eq 'TAG' ) ){ 
       print $saveto."\n" if ($saveto) ; 
       $saveto=''; 
       $save=0; 
       } 
  elsif ($save) { $saveto.=' '.$line;); } 
  }
print $saveto."\n" if ($saveto) ;
[download]

Im not gonna harp about lexical filehandles or 3 arg opens, somebody else can.

@lines = <FILE1>; "ate" the whole file at once. there was nothing left for the rest of your loop that wouldnt work anyway.

There better be a X2 or BEGIN TAG before a X3 or it will eat it too.