Working, tested code:

#!/usr/bin/perl
use strict;
use warnings 'all';

use Win32::OLE;
use Win32::OLE::Const 'Microsoft Internet Controls';
use Win32::OLE::Variant;

# From Vc7/PlatformSDK/Include/MsHtmHst.h :
use constant PRINT_DONTBOTHERUSER    => 0x01;
use constant PRINT_WAITFORCOMPLETION => 0x02;

my $file = "C:\\test.html";
my $do_not_prompt = 0;

my $nCmdID      = OLECMDID_PRINT;
my $nCmdExecOpt = OLECMDEXECOPT_PROMPTUSER;
my $pvaIn       = PRINT_WAITFORCOMPLETION;
my $pvaOut      = 0;

if ($do_not_prompt) {
  $nCmdExecOpt  = OLECMDEXECOPT_DONTPROMPTUSER;
  $pvaIn       |= PRINT_DONTBOTHERUSER;
}

my $IE = Win32::OLE->new('InternetExplorer.Application') or die;
#$IE->{'Visible'} = 1;
$IE->Navigate( $file );
sleep 1 while $IE->{Busy};

$IE->ExecWB($nCmdID, $nCmdExecOpt, Variant(VT_I2,$pvaIn), $pvaOut);

$IE->Quit();
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Comments:

1. The $do_not_prompt option was included to allow for better exploration of the timing issues.
2. The error  "Trying to revoke a drop target that has not been registered" is indeed caused by MSIE not being ready. The QueryStatusWB method that you mentioned might work fine. I originally used sleep 1 while $IE->{ReadyState} != READYSTATE_COMPLETE, but found that the $IE->{Busy} method also worked, and was more concise.
3. None of those methods work to delay the end of the ExecWB call, because the printing happens in the background. If we don't PRINT_WAITFORCOMPLETION, (or sleep($hope_it_never_takes_longer_than_this)), then MSIE will often exit before the printing begins.
4. When ExecWB is called with OLECMDID_PRINT, the third parameter is overloaded by Variant type (VT_BSTR, VT_I2, or VT_ARRAY). If I just passed $pvaIn directly, then it would have been converted to the default numeric Variant type (VT_I4), which would then be disregarded! I had to build the specific Variant that OLECMDID_PRINT wants, to make it work. See the MSDN page for details.
5. In your second section of code, ${ $ie_const }{ 'OLECMDID_PROMPTUSER' } will not be defined.
   The constant you wanted was OLECMDEXECOPT_PROMPTUSER. If you import the constants, as I have done in my code, then this kind of error will be caught at compile time. As a guideline, I import when using constants as individual values, and only use the Win32::OLE::Const->Load() form when iterating through sets of constants.

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