The output is buffered and you didn't wait long enough. (4096b / 25b/msg * 5s/msg ≤ 164s)

Buffering can be turned off:

use strict;
use warnings;

use IO::Handle qw( );

my $log = "/tmp/log.txt";

open my $LOG, ">", $log or die "$!\n";
$LOG->autoflush(1);

while ( 1 ) {
     my $time = localtime() ;
     print $LOG "$time\n";
     print "$time\n";
}

close $LOG;
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