Can't use string ("00000140") as a symbol ref while "strict refs" in use at ./partition_file.pl line 26, <$FH> line 6.

The code you posted doesn't produce this error message, it most likely stems from an earlier version of your program, in which you e.g. used $OUT overall instead of $NEW. In this case, in line 26, you would be trying to use a string as a filehandle.

Are you sure that your "mrule" tokens are grouped? If after mrule=150 some line containing mrule=140 is found further down in the source file, you overwrite the file 00000140.log with that line.

To avoid that, open a file for every mrule token and store the open filehandles in a hash:

my %fh;
while ( my $line = <$FH> ) {
    if ( $line =~ /mrule=([0-9]+)/){
        my $mrule = sprintf("%08d.log", $1);
        if ( ! $fh{$mrule} ) {
            open my $fh, '>', $mrule or die "Can't write to '$mrule': 
+$!\n";
            $fh{$mrule} = $fh;
        }
        print {$fh{$mrule}} $line;
    }
}
close $fh{$_} for keys %fh; # close all files
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Note the extra braces in the print statement around $fh{$mrule}. These are necessary to disambiguate $fh{$mrule} as being a filehandle rather than part of the LIST for print.