While I like this idea, a couple of minor comments. First, there's an awful lot of shifting going on. For things like this, the topicaliser, $_, is very useful. So you'd be able to set things up as:

checkdir('/usr/tmp', sub { -w $_ });
checkdir('/usr/tmp', sub { -r $_ });
checkdir('/usr/tmp', sub { -r $_ and -w $_ });
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At this point, I want to point out my second minor comment. This is what _ is for.

# now it's:
checkdir('/usr/tmp', sub { -r $_ and -w _ });
checkdir('/usr/tmp', sub { !/\.\./ });
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Again, my next minor comment: I'm not sure what this is supposed to be checking. I'm guessing you're looking for a directory named ".." somewhere in there. However, I'm guessing that "/usr/blah..foo" should be permitted? Regexes are powerful, but with power comes great respon... er, complexity ;-) Rather than use a regex, I would actually suggest doing exactly what a human would do: split the path elements up, and then look that none of them are equal to "..". If that's what you really want. (On unix, I claim that this is almost never what you want, due to symlinks.)

use List::MoreUtils qw/none/;
use File::Spec;

checkdir('/usr/tmp', sub {
  none { $_ eq '..' } File::Spec->splitdir($_)
});
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With those minor comments, here's the minor change to checkDir:

sub checkDir {
  my ($path, $perm) = @_;
  ### Check if output dir exists
  if (! defined $path) {
    LOG ("Output path not defined; using temp instead...", 1);
    $path = '/var/tmp/';
   }
  if (! -d $path) {
    die LOG ("$path is not a valid directory...", 0);
  }
  if (! do { local $_ = $path; $perm->() }) {
    die LOG ("Test failed on $path, not able to use it.", 0);
  }
  LOG ("Using $path as output directory...", 2);
}
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A bit more convoluted internally, but I think a much more perlish exterior.