Works for me...

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

my $line = "X\tN";
my @tags = split /\t/, $line;
my @Definition=("A", "NN", "B", "NNS", "YN", "ZX");
my @out = grep($_ =~ /^$tags[1]/, @Definition);
print Dumper(\@out);

__END__

$VAR1 = [
          'NN',
          'NNS'
        ];
[download]

Could you show an SSCCE where it fails for you? Just an idea, you might want to check for stray whitespace in @tags (see Basic debugging checklist).

By the way, the match against $_ is implicit, and you probably want to use \Q...\E to have regex metacharacters match literally (see e.g. Mind the meta!). Also, I prefer the {BLOCK} form of grep - I would have written grep {/^\Q$tags[1]\E/} @Definition.

Minor ninja edits.

Update: If you instead mean that you want to check if $tags[1] begins with one of the strings in @Definition, then one way to do that is described in Building Regex Alternations Dynamically:

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

my @Definition=("NN", "NNS");
my ($defs) = map { qr/$_/ } join '|', map {quotemeta}
    sort { length $b <=> length $a } @Definition;

for my $str ( "ABC", "NXY", "NNF", "NNSX" ) {
    print "$str: ", $str=~/^$defs/ ? "yes!\n" : "no\n";
}

__END__

ABC: no
NXY: no
NNF: yes!
NNSX: yes!
[download]

(Although ("NN", "NNS") is kind of redundant, ("NN") is enough.)