Read line by line is not a good idea. You should use read() to read in a chunk, then do one <> to make sure the small files end at line ends. (I tested the demo with a big log file, the performance is very good.)

use strict;
use warnings;

open (FH, "<message_log01") or die "Could not open source file. $!";

my $i = 0;

while (1) {
    my $chunk;
    print "process part $i\n";
    open(OUT, ">part$i.log") or die "Could not open destination file";
    $i ++;
    if (!eof(FH)) {
        read(FH, $chunk, 1000000);
        print OUT $chunk;
    } 
    if (!eof(FH)) {
        $chunk = <FH>;
        print OUT $chunk;
    }
    close(OUT);
    last if eof(FH);
}
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