Firstly, grep is excessive. Since your keys are upper-case, simply use uc to upper-case things before matching.

#!/usr/bin/perl
use strict;
use warnings;

my @words;
while (<STDIN>) { push @words, split /\s/ }
my %wordscore = scrabble_score(@words);
for (sort keys %wordscore) {
    print qq/"$_" = $wordscore{$_} points\n/;
} 


sub scrabble_score {
    my %value = ( A => 1,  F => 4,  K => 5,  P => 3,   U => 1,  Z => 1
+0,
            B => 2,  G => 2,  L => 1,  Q => 10,  V => 4,
            C => 3,  H => 4,  M => 3,  R => 1,   W => 4,
            D => 2,  I => 1,  N => 1,  S => 1,   X => 8,
            E => 1,  J => 8,  O => 1,  T => 1,   Y => 4
    );
    
    my %score;
    for (@_) {
        my @letter = split "", uc $_;
        my $word = $_;
        $score{$word} = 0;
        for (@letter) { $score{$word}+=$value{$_} }
    }
    
    return %score;
}
[download]

Note that it reads STDIN until EOF -- Ctrl-D on Unix/Linux and Ctrl-Z on Windows (followed by Enter on some of both types of systems). Or you can pipe in a word list... but you needn't have one word per line, as long as you use space between them.

Also, because of how hashes function, it will only print one instance of each word.

By the way, you aren't checking to make sure the word is possible in Scrabble: there are a limited quantity of each letter... :)