You can achieve what you want like so:

use warnings;
use strict;
use Data::Dumper;

my $device_id = {};
my $id        = "";

while (<DATA>) {
    chomp;
    if (m/Logical.+=(.+?)$/) {
        $id = $1;
    }
    else {
        if (m/.+?\s+?(c.+?)\s+?.+?$/) {
            push @{ $device_id->{$id} }, $1;
        }
    }
}

print Dumper($device_id);

__DATA__
### HEADING OF RECORD 1 ####
Logical device ID=08E1
LINE_THAT_DOES_NOT_BOTHER_ME
ANOTHER_LINE_THAT_DOES_NOT_BOTHER_ME
29 8/0/2/1/0.18.152.0.0.6.1  c29t6d1   FA  5eA
30 8/0/3/1/0.17.152.0.0.6.1  c30t6d1   FA 12e
31 8/0/8/1/0.17.150.0.0.6.1  c31t6d1   FA 10eA
32 8/0/9/1/0.18.150.0.0.6.1  c32t6d1   FA 11eA

### HEADING OF RECORD 2 ####
Logical device ID=08E2
LINE_THAT_DOES_NOT_BOTHER_ME
ANOTHER_LINE_THAT_DOES_NOT_BOTHER_ME
29 8/0/2/1/0.18.152.0.0.4.1  c29t4d1   FA  5eA
30 8/0/3/1/0.17.152.0.0.4.1  c30t4d1   FA 12eA
31 8/0/8/1/0.17.150.0.0.4.1  c31t4d1   FA 10eA
32 8/0/9/1/0.18.150.0.0.4.1  c32t4d1   FA 11eA

### HEADING OF RECORD 3 ####
(...)
[download]

output:
$VAR1 = {
          '08E2' => [
                      'c29t4d1',
                      'c30t4d1',
                      'c31t4d1',
                      'c32t4d1'
                    ],
          '08E1' => [
                      'c29t6d1',
                      'c30t6d1',
                      'c31t6d1',
                      'c32t6d1'
                    ]
        };
[download]

Please, note test your match regexes.
Also Check perldoc perldsc
Hope this helps