It's not clear what output you expect. To search for overlapping sequences, you can change the second group from non-grouping to a look-behind: 
$sequence =~ /(ATG.*?(?<=TAA|TAG|TGA))/g

[download]

to get 

ATGGTTTCTCCCATCTCTCCATCGGCATAA
ATGA

[download]

It still extracts the shortest possible sequence for each starting point (so we lost the second output).

Update: It's possible to get all the sequences without experimental regex features and depending on the return value of print like here. 

my @from;
my $pos = -1;
push @from, $pos while -1 != ($pos = index $sequence, 'ATG', $pos + 1)
+;

my @to;
for my $end (qw( TAA TAG TGA )) {
    $pos = -1;
    push @to, $pos + 3 while -1 != ($pos = index $sequence, $end, $pos
+ + 1);
}

for my $f (@from) {
    for my $t (@to) {
        say substr $sequence, $f, $t - $f if $t > $f;
    }
}
__END__
Output:

ATGGTTTCTCCCATCTCTCCATCGGCATAA
ATGGTTTCTCCCATCTCTCCATCGGCATAAAAATACAGAATGATCTAA
ATGGTTTCTCCCATCTCTCCATCGGCATAAAAATACAGAATGA
ATGATCTAA
ATGA

[download]
لսႽ† ᥲᥒ⚪⟊Ⴙᘓᖇ Ꮅᘓᖇ⎱ Ⴙᥲ𝇋ƙᘓᖇ

In reply to Re: Print A Sequence with Start codon and different Stop Codon by choroba
in thread Print A Sequence with Start codon and different Stop Codon by PerlKc

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