I know I went a little overboard with the comments but I wanted to try to make things more clear.

I understand the feeling, but if your variable and function names are meaningful (and you use functions), you need less comments

I'll give you an example, this part of your code

          
              for my$i(0..($replication_rounds-1))
            {
                 for my $j (0..(@{$allgen{$i}}))
                 {
                    my $month=(int($i/30)+1); # the counter divided by
+ 30 will give a decimal, the int rounds it down... therefore .2 -->0 
+(then the +1) yields a january
                    if ($month<10){$month="0".$month;} ## too add a ze
+ro which mat lab reads easier so 09 instead of 9 for sept
                    my $date="000".(int($month/12))."/".$month."/".($i
+-(($month-1)*30)); # year is set at 0000 for now.... change 000 to 20
+0 or whatever to make it a specific year
                  
                    print "\>RandomHA ". $date ." "."\n" . $allgen{$i}
+[$j-1] . "\n";
                  
                  # to print a file automatically in fasta format
                    #open(FILE,">>Rand_HA_.txt");
                    #print FILE "\>RandomHA ". $date ." " ."\n" . $all
+gen{$i}[$j-1] . "\n";
                    #close(FILE);
                }
            }
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for my $i ( 0 .. ( $replication_rounds - 1 ) ) {
    for my $j ( 0 .. ( @{ $allgen{$i} } ) ) {
# the counter divided by 30 will give a decimal,
#~ the int rounds it down...
#~ therefore .2 -->0 (then the +1) yields a january
        my $month = ( int( $i / 30 ) + 1 );
        if ( $month < 10 ) {
            $month = "0" . $month;
        }
## too add a zero which mat lab reads easier so 09 instead of 9 for se
+pt
        my $date = "000"
          . ( int( $month / 12 ) ) . "/"
          . $month . "/"
          . ( $i - ( ( $month - 1 ) * 30 ) ) ;
# year is set at 0000 for now....
# change 000 to 200 or whatever to make it a specific year

        print "\>RandomHA " . $date . " " . "\n" . $allgen{$i}[ $j - 1
+ ] . "\n";

    } ## end for my $j ( 0 .. ( @{ $allgen...}))
} ## end for my $i ( 0 .. ( $replication_rounds...))
###################
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for my $i ( 0 .. ( $replication_rounds - 1 ) ) {
    for my $j ( 0 .. ( @{ $allgen{$i} } ) ) {
        my $date = BlahIjMeSomeDate( 200, $i, $j );
        print "\>RandomHA $date \n" . $allgen{$i}[ $j - 1 ] ."\n";
    }
}
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for my $i ( 0 .. ( $replication_rounds - 1 ) ) {
    for my $j ( 0 .. $#{ $allgen{$i} }  ) {
        my $date = BlahIjMeSomeDate( 200, $i, $j );
        print "\>RandomHA $date \n" . $allgen{$i}[ $j ] ."\n";
    }
}
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See, doesn't that make more sense?

Similarly foreach my $nucleotide (@nucleotide_array) { ... I would rewrite as

foreach my $nucleotide (@nucleotide_array) {
    chomp $nucleotide;
    my $score = ( int rand(100000) ) / 100000;
    ModifyNucleotideBasedOnScore( \$nucleotide, $score );
}

sub ModifyNucleotideBasedOnScore {
   my( $nucleotide, $score ) = @_;
   ...
   $$nucleotide =~ s/A/T/g;
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Now I'm not sure about this bioinformatics thing, but if $nucleotide is one character long, you can speed up the program by NOT using substitution operator, use assignment.

I let your program run for 10min and it grew to about 140mb in memory, at which point I killed it

Changing =~ s/... to straight assignment, shaved off a few minutes off (it reached ~140mb in memory in about 8 min)

To make sure you're squeezing the most speed out of the shuffle make sure you're using the XS version of List::Util (see About List::Util's pure Perl shuffle())

Since your program seems to be CPU/Memory bound and not IO bound, you could probably speed it up with threads, something like Re: Thread::Pool shutdown dies after abort or

my $maxNumberOfThreads = 5;
my $threadCount = 0;
for ( $i = 1 ; $i < $replication_rounds ; $i++ ) {
    my $thr = threads->create(
        {'context' => 'list'},
        \&ReturnSomeFilteredNewGen,
    );
    $threadCount++;
    next if $threadCount < $maxNumberOfThreads ;
    JoinJoinables(\%allgen);
}
JoinJoinables(\%allgen);


sub JoinJoinables {
    my( $allgen ) = @_;
    while( my @joinable = threads->list(threads::joinable) ){
        for my $thr( @joinable ){
            my( $i, @filtered_new_gen ) = $thr->join();
            $$allgen{$i} = \@filtered_new_gen;
        }
    }
}
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