Then you should definitely use a module, such as one of those recommended by aitap above:

#! perl
use strict;
use warnings;
use autodie;
use Capture::Tiny qw( tee_stdout );

my %test_hash = ( Foo => 1, Bar => 1, Baz => 1, Nix => 0 );

foreach (keys %test_hash)
{
    if ($test_hash{$_} == 1)
    {
        my $logfile = $_ . '_log.txt';
        print "the logfile is $logfile\n";

        require "$_.pm";
        my ($stdout, @result) = tee_stdout { $_->$_() };

        open(my $out, '>', $logfile);
        print $out $stdout;
        close($out);
    }
}
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++aitap for recommending Capture::Tiny! I already had it on my system (it came with Strawberry Perl), but I hadn’t come across it before. Brilliant!