Here's a reformed version. Your main problem seemed to be the extra increment of $i down at the bottom. There is no reason to do this with a loop if you already know the position so I removed it and inlined one of the subs. Give it a shot.

use warnings;
use strict;

my $seq = 'AAAAAAAAAAAAAAAAAAAAAAAAAAAAAA';
print "The DNA sequence is : $seq\n";

mutate($seq);

exit;

sub mutate {
    my $dna = shift;
    my $position = int(rand(length($dna)));

    print " mutation at  base position ". ($position+1) . "\n";

    my $nucleotide;
    do {
        $nucleotide = randombase();
    } until ( $nucleotide ne substr($dna,$position,1) );

    substr($dna, $position, 1, $nucleotide);
    print "$dna\n";
}

sub randombase {
    my ( @nucleo ) = qw ( A T C G );
    return $nucleo[rand(@nucleo)];
}
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