I just changed your regular expression to match 0 to 2 control chars and replace them with nothing otherwise I did not touch the code.
[:cntrl:] is the key to matching any control chars.

#!/usr/bin/perl -w
use strict;
my ($infile, $outfile, $line);

$infile = "c:/perldev/restore.txt";
$outfile = "c:/perldev/fixed_restore.txt";

open (INFILE, $infile) || die "Can't open $infile! $!";
open (OUTFILE, ">$outfile) || die "Can't open $outfile! $!";

while (<INFILE>) {
    $line = $_;
    $line =~ s/^[:cntrl:]{0,2}//;
    print OUTFILE $line;
}
close (INFILE);
close (OUTFILE);
[download]

-monkfish (the fishy monk)

Edit fixed error with \C[