Two suggestions

1. You do not need the for loop at all, you are merely counting to 30 and doing something once when you reach a magic number. This is particularly true if the starting dna sequence is greater than 30 bases.
2. A look up table would be faster then an "until we get a different answer", this is particularly true if you are going to mutate your starting sequence repeatedly.

Try this for example

#!/usr/bin/perl 

use strict;
use warnings;

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

print "$seq \n";
mutate ($seq);
exit;

sub mutate{
        my($dna) = shift;
        my $position = randposition($dna);
        print " mutation at  base position ". ($position+1) . "\n";
        my $nucleotide = randombase(substr($dna,$position,1));
        substr($dna, $position, 1, $nucleotide);
        print "$dna\n";
}

sub randposition{
        my($sequence) = @_;
        return int rand length($sequence);
}

sub randombase{
        my $base=shift;
        my(%nucleo)=(   'A'=>['T','C','G'],
                        'T'=>['A','C','G'],
                        'C'=>['A','T','G'],
                        'G'=>['A','T','G'],
                    );
        return $nucleo{$base}[rand(@{$nucleo{$base}})];
}
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