I would use a loop to parse the input...

use strict;
use warnings;

my $chunk = 0;
my $flag  = 0;

open my $fh, '<', "data.txt";
while (my $line = <$fh>) {
    chomp $line;
    my ($code, $state) = split / +/, $line;
    if ($flag and $state eq 'False') {
        print "\n#Chunk $chunk\n";
        $chunk++;
    }
    print "$line\n";
    $flag = 1;
}
[download]

C:\Users\user\Perl>perl test.pl
GO:0002366       False
GO:0002446       True
GO:0002275       True
GO:0043312       True

#Chunk 0
GO:0006733       False

#Chunk 1
GO:0019674       False

#Chunk 2
GO:0043588       False

#Chunk 3
GO:0055065       False
GO:0055080       True
[download]

UPDATE: I've just noticed you want #Chunk 0 at the top so no $flag is needed to suppress it...

use strict;
use warnings;

my $chunk = 0;

open my $fh, '<', "data.txt";
while (my $line = <$fh>) {
    chomp $line;
    my ($code, $state) = split / +/, $line;
    if ($state eq 'False') {
        print "\n#Chunk $chunk\n";
        $chunk++;
    }
    print "$line\n";
}
[download]

Output:

C:\Users\user\Perl>perl test.pl

#Chunk 0
GO:0002366       False
GO:0002446       True
GO:0002275       True
GO:0043312       True

#Chunk 1
GO:0006733       False

#Chunk 2
GO:0019674       False

#Chunk 3
GO:0043588       False

#Chunk 4
GO:0055065       False
GO:0055080       True

C:\Users\user\Perl>
[download]