To follow your general program logic, but with the help of File::Find::Rule and File::DosGlob, the latter of which is a core module:

#!perl
use strict;
use warnings
use autodie; # errors with open, chdir, etc are now fatal

use File::Find::Rule;
use File::DosGlob 'glob';

my $search = shift || 'c:/Documents and Settings/user/Desktop/newbatch
+2' ;
my $outfile = 'c:/Documents and Settings/user/Desktop/newbatch/log.txt
+';

open my $outf, '>', $outfile;

for my $dir (File::Find::Rule->dir->in($search)) {
    for my $file ( <*>) {     
        my $path = "$dir\\$_";
        print " $path\n";
        if (-f $path) {
            if ($file ne 'empty.txt' and $file ne 'sample.txt') {
                unless (-s $path) {
                    print $outf "Warning: $path has size ZERO\n";
                }
            }
        }
    }
}
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OR, to be more concise:

#!perl
use strict;
use warnings;
use autodie;

use File::Find::Rule;
use File::DosGlob 'glob';

my ($search) = @ARGV or die "need a directory to search";

sub found {
    my ($name, $path, $full) = @_;
    warn "Warning: $full has size ZERO" # outputs to STDERR
        unless $name ~~ ['empty.txt', 'sample.txt']; # Perl v5.10+
#        unless $name eq 'empty.txt' or $name eq 'sample.txt';
}

File::Find::Rule->file->empty->exec(\&found)->in($search);
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1)Always use strict; and use warnings;. Even on Windows, #!/usr/bin/perl -w is equivalent to use warnings; although the path to Perl is ignored.
2)File::Find will visit all files and directories underneath the starting directory. There is no need to save the directories and then re-open them only to read the files names yet again. All of those file names and the directory have already been visited!
3)If what you need to do to the file is not complicated, do it right in the File::Find subroutine - No need to save it's path and come back.
4) Please pay more attention to the indenting of your code - it is important.
5) I think the code below is essentially equivalent to what your code does, reports zero length files that are found.
6) I was not able to understand your question about what you want to do other than this. "But instead of simply generating a log file..." Please try a different explanation.

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

use File::Find;
use File::Basename;

my $start_dir = "C:/temp"; 
my $log_file = 'C:/temp/logfile.txt';
open (my $log, '>', $log_file) or die "cannot open $log_file for write
+";

find (\&process_file, $start_dir);

sub process_file
{
    return unless -f;  # do not process directories, 
                       # only "real files"
    my $full_path_name = $File::Find::name;
    print "$full_path_name\n";  
    
    #other ways of writing the "if", but extending your code
    
    if ( -s == 0 and ($_ ne 'empty.txt') and ($_ ne 'sample.txt') and 
                     ($full_path_name ne $log_file) 
       )
    {
       print $log "Warning: $full_path_name has size ZERO\n";
    }    
}
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