Data::Compare can do this. Here's one way:

use warnings;
use strict;

use Data::Compare;
use Data::Dumper;

my $arefs = [
    [
      {
        BIRTH        => "11/04/2014",
        CODE => 4,
        NAME      => "JOHN D.",
        NUMBER      => 1234,
      },
    ],
   [
      {
        BIRTH        => "11/04/2014",
        CODE => 4,
        NAME      => "JOHN D.",
        NUMBER      => 1234,
      },
    ],
    [
      {
        BIRTH        => "11/04/2014",
        CODE => 4,
        NAME      => "Mike D",
        NUMBER      => 5555,
      },
    ],
];

my $no_dups_aref;

while (my $aref = shift @$arefs){
    my $dup = 0;
    for (@$arefs){
        $dup = Compare($aref, $_);
        last if $dup;
    }
    next if $dup;
    push @$no_dups_aref, $aref;
}

print Dumper @$no_dups_aref;
[download]

However, I'd be seriously inclined to change your data structure. There's no need to have nested arrays if all you have in each one only has a single hash. I'd dumb it down to a simple AoH:

use warnings;
use strict;

use Data::Compare;
use Data::Dumper;

my $hrefs = [
      {
        BIRTH        => "11/04/2014",
        CODE => 4,
        NAME      => "JOHN D.",
        NUMBER      => 1234,
      },
      {
        BIRTH        => "11/04/2014",
        CODE => 4,
        NAME      => "JOHN D.",
        NUMBER      => 1234,
      },
      {
        BIRTH        => "11/04/2014",
        CODE => 4,
        NAME      => "Mike D",
        NUMBER      => 5555,
      },
];

my $no_dups_aref;

while (my $href = shift @$hrefs){
    my $dup = 0;
    for (@$hrefs){
        $dup = Compare($href, $_);
        last if $dup;
    }
    next if $dup;
    push @$no_dups_aref, $href;
}

print Dumper @$no_dups_aref;
[download]