OK, thanks for your help. Below is what I came up with. However, it only works if I only recently entered my password with sudo from the command line. If it's been a while since logging in with sudo, I get this output from the script:

STDOUT:
STDERR: Password:

return:
STDOUT:
STDERR: Password:

return:
STDOUT:
STDERR: Password:

return:
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So it doesn't look like the Sudo module is submitting the password properly. I'm on Mac.

UPDATE! I fixed it by putting in a newline character after the password. It works! Thanks so much for your help. Code below modified to reflect the fix.

use Modern::Perl;
use Sudo;
use File::Copy;

my $tmp_file = '/Users/my_home/tmp/hosts';
my $target = '/etc/hosts';
my $user = 'root';
my $pass = "hardtoremember\n";
my $sudo = '/usr/bin/sudo';

my $uid = (stat $target)[4];
my $file_user = (getpwuid $uid)[0];

my $gid = (stat $target)[5];
my $file_group = (getgrgid $gid)[0];

my $mode = (stat($target))[2];
my $perms = sprintf("%04o", $mode & 07777);

# copy file to tmp file
unlink $tmp_file;
copy ('/etc/hosts', $tmp_file);

# append tmp file
system ("echo '10.0.1.17 site.com' >> $tmp_file");

# change permissions
my $su = create_cmd('/bin/chmod', "$perms $tmp_file");
check_output($su->sudo_run);

# change file ownership
$su = create_cmd('/usr/sbin/chown', "$file_user:$file_group $tmp_file"
+);
check_output($su->sudo_run);

# copy modified file back
$su = create_cmd('/bin/cp', "$tmp_file $target");
check_output($su->sudo_run);

# cleanup
unlink $tmp_file;

sub check_output {
  my $result = shift;
  if (exists($result->{error})) {
    print $result->{error};
  } else {
         printf "STDOUT: %s\n",$result->{stdout};
         printf "STDERR: %s\n",$result->{stderr};
         printf "return: %s\n",$result->{rc};

  }
}

sub create_cmd {
  my $cmd = shift;
  my $args = shift;
  return Sudo->new(
    { 
      sudo => $sudo,
      username => $user,
      password => $pass,
      program => $cmd,
      program_args => $args,
    }
  );
}
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