$_=~s/$i//; is converting some members of @prime into the zero-length string. The zero-length string gets interpreted as zero in a numerical context. It could be fixed by skipping blank lines as seen below.

I took the liberty of making a few optimizations:

• I simplified your if-prime expression.
• I removed the non-special case of "2".
• I simplified the loop and the regexp into a simple assignment.
• I exit the loop early (via last) when appropriate.
• I pluralized @prime to @primes.

my @primes;
for my $i (2..100) {
    $primes[$i] = $i;
    foreach (@primes) {
        if ($_ && $i % $_ == 0) {
            $primes[$i] = '';
            last;
        }
    }
}

foreach (@primes) {
    next unless $_;
    print "$_\n";
}
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You could also avoid adding it in the first place, as shown in the following snippet.

my @primes;
NUMBER:
for my $i (2..100) {
    foreach (@primes) {
        if ($i % $_ == 0) {
           # Divides evenly, so not prime.
           next $NUMBER;
        }
    }
    push @primes, $i;
}

foreach (@primes) {
    print "$_\n";
}
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The disadvantage of the latter is that you can't do if ($prime[$i]) to determine if $i is prime. The advantage is smaller memory usage and a (presumably) small increase in speed.

Update: Cleaned up/Expanded the text.