A few things with that... while (<$FILE>) {
would assume you had a variable that is a glob of a filehandle. I'm not sure why you'd want to do that, just a caveat.

elif should be elsif

and your code also assumes both a single line read (in the /^\d/ case) and a multi-line read (in the /^\s+ID/ case), unless I'm reading that incorrectly.

Here's what I came up with:

use Data::Dumper;
use strict;

my $hash = {};

my @files = ( "LOG1", "LOG2", "LOG3" );

foreach my $file ( @files ) {
  open FILE, "<", $file or die "Unable to open $file\n$!\n";
   
  my ( $date, $id, $timestamp ) = qw( Unknown "" "" );
  while (<FILE>) {
     chomp;
     if (/^\d/){ 
        $date = $_;
     } elsif (/^\s+ID\s*=\s*(.*)$/) {
        $id = $1;   
     } elsif ( /^\s+TIMESTAMP\s*=\s*(.*)$/ ) {
        $timestamp = $1;
     } elsif ( /^\s+COUNT_\d+\s*=\s*(.*)$/ ) {
        $hash->{$date}->{$id}->{$timestamp} += $1;
     }
  }
}

print Dumper($hash), "\n";
[download]

which would give you

$VAR1 = {
          '20090619' => {
                          '4' => {
                                   '1244127600' => '20'
                                 },
                          '1' => {
                                   '1244127600' => '20'
                                 },
                          '3' => {
                                   '1244127600' => '10'
                                 },
                          '0' => {
                                   '1244127600' => '20'
                                 },
                          '2' => {
                                   '1244127600' => '10'
                                 }
                        }
        };
[download]

Assumptions are that the log files are all in the order specified, the actual COUNT_X values are immaterial, etc.