The problem is evaluating e for each iteration of the while loop. If you make the ref assignment outside the loop things work as expected:

use warnings;
use strict;

my %env= %{e ()};

$|=1;
while (my ($var, $val) = each %env) {
      print "(hash) $var=$val\n";
}

my $ref = e ();
while (my ($var,$val) = each %$ref) {
      print "(ref)  $var=$val\n";
}

sub e {
  my %result=(A=>1,B=>2);
  wantarray ? each %result : \%result;
}
[download]

Prints:

(hash) A=1
(hash) B=2
(ref)  A=1
(ref)  B=2
[download]

As an aside: prototyping subs in Perl is only very occasionally required and should not be done as a matter of course. Likewise, calling subs using & is magical and should only be used when required.