This doesn't handle this case:

// We do not use /*-style comments
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It doesn't even handle the case:

// We don't use old C-style comments
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because it tries to find the closing single quote to match the apostrophe in "don't". You simply have to parse //-style comments for such a tool.

/"([^"]|\\­")*"/
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This doesn't handle "\\". Also note that it will fail for strings of 32K characters, which is why I prefer to add the + in "([^"\\]+|\\­.)*".

Why don't you factor out m{((?! \*/ | /\* ).)+}sx into its own rule so you don't have to repeat that regex three times and so you can assign a descriptive name to it to aid understanding?

m{((?!/\*|­"|').)+}s could be replaced by [^/"'] and /(?![*]), which is more to my taste but YMMV.

And I'd probably do this all with simpler regexes and a simple state machine instead of resorting to Parse::RecDescent (not that my result will be simpler code in total). Note that I even avoid having to slurp the entire input into a single string.

#!/usr/bin/perl -w
use strict;

$|= 1;              # Useful for ad-hoc testing
my $canNest= 1;     # Whether /*-style comments can be nested

my $depth= 0;
my $output= "";
while( <DATA> ) {
    while(  ! m[\G\z]gc  ) {
        while(  $depth  &&  m[/[*]|[*]/]gc  ) {
            if(  "/" eq substr( $_, $-[0], 1 )  ) {
                $depth++;
            } elsif(  $canNest  ) {
                $depth--;
            } else {
                $depth= 0;
            }
        }
        last   if  $depth;
        if( m[
                \G
                (?:
                    [^'"/]+
                  | ' (?: [^'\\]+ | \\. )* '
                  | " (?: [^"\\]+ | \\. )* "
                  | /(?![/*])
                )+
            ]xgc
        ) {
            $output .= substr( $_, $-[0], $+[0] - $-[0] );
        } elsif(  m[\G//.*]gc  ) {
            # skip C++ comments
        } elsif(  m[\G/[*]]gc  ) {
            $depth++;
        } elsif(  m[\G['"]]gc  ) {
            warn "Ignoring unclosed quote: $_";
        } else {
            die $_, ' ' x pos($_), "^\nCouldn't be parsed";
        }
    }
    print $output;
    $output= "";
}
warn "$depth unclosed /*-comments\n"   if  $depth;
__END__
#include "StdAfx.h" // Tail comment
#include "Utility\perftime.h"
#pragma hdrstop

/* Comment before MACRO */
/* Comment /* and nested comment */ lines */

#define MACRO 10\
              + 3 // Multi line macro with comment

#define __DEBUG /* comment */ 1
#define STRING 'This is a string' /* comment */
#define BACKSLASH '\\'
#define COMMENT "/* comment in \"a\" string */"

// c++ comment line
/* Comment at start for a number of lines */
/* multi-line comment
/* nested */
block */
// cpp block

char PerfTimer::Buf[64];
// Don't use contractions
// /*-style comment below over multiple lines:
test/*ing how newlines work
when a comment spans lines, does it st*/ing?
total/*divide*//count//*comment
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Produces

#include "StdAfx.h"
#include "Utility\perftime.h"
#pragma hdrstop




#define MACRO 10\
              + 3

#define __DEBUG  1
#define STRING 'This is a string'
#define BACKSLASH '\\'
#define COMMENT "/* comment in \"a\" string */"






char PerfTimer::Buf[64];


testing?
total/count
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(Very minor updates applied.)