Before any other debugging, other than ensuring the code compiles, you should be checking that open() succeeded (unless you're already using autodie):

my $dst = "C:\\Users\\Administrator\\Desktop\\Perl_Scripts\\8312032_Se
+curity.evtx";
my $src = "C:\\Users\\Administrator\\Desktop\\Perl_Scripts\\8312032_Sy
+stem.evtx";

open (my $dsth, ">>", $dst)
  or die qq(can't open "$dst" for appending: $!);
open (my $srch, "<", $src)
  or die qq(can't open "$src" for reading: $!);

print {$dsth} <$srch>; 

close ($srch);
close ($dsth);
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or

use autodie;
{
    open (my $dsth, ">>", $dst);
    open (my $srch, "<", $src);
    print {$dsth} <$srch>; 
}
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That last block could be any type of block providing lexical scope. The print {$filehandle} LIST; construct makes it more obvious that you're printing to a filehandle object, rather than missing a comma in your arguments list, to both the programmer and to Perl. In list context, the diamond operator returns all lines from a file. You don't need to escape colons or underscores in a string or a regular expression.

For easy of readability, Perl can handle Windows paths with forward slashes (actually, Windows itself handles forward-slash paths fine, which is what Perl sends to the system -- its Windows built-in applications which force only backslash paths on you). So to prevent forgetting to backslash a directory separator and to minimize length of paths strings, you could represent them as:

my $dst = "C:/Users/Administrator/Desktop/Perl_Scripts/8312032_Securit
+y.evtx";
my $src = "C:/Users/Administrator/Desktop/Perl_Scripts/8312032_System.
+evtx";
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