The hash approach is better for that, but I went ahead and re-implemented the regexp approach again anyway, just in case someone is interested in a pattern matching solution rather than an 'equal key' solution.

If you wanted to continue with the regexp approach, this solution will count the number of duplicate words. I modified the RE a little so that it would count "Antony Antony Antony" as two duplicates (Antony is repeated twice after the original). "Antony Antony Hank Antony Hank Mark" would count 3: Antony has two repeats, and Hank has one.

use warnings;
use strict;

my $string='LOCAL    Antony    17    Antony    23    1569';

my $count;
$count++ while
    $string =~ m/
                  \b([[:alpha:]]+)\b
                  (?=.*?\b\1\b)
                /xg;

print $count, "\n";
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$hash{$_}++ foreach(split /\t/,$string);
foreach (keys (%hash)) {print $_."\n" if ($hash{$_} > 1 && /^\D+$/)}
[download]

I don't want to know what the dulicates are, I only want to know if there are any...

I'm not sure it's possible to know the latter without also knowing the former. At any rate, any of the solutions shown so far will do the job; just ignore what the actual duplicate values are. For example, my solution can be modified very slightly:

use Scalar::Util qw( looks_like_number );

my %h;
$h{$_}++ for split /\t/, $string;

my $there_are_duplicates =
    grep { $h{$_}>1 and !looks_like_number($_) }
    sort keys %h;
[download]

(This exploits the fact that grep returns the list of matching values in list context, and returns the number of matches in scalar context.)