as a suggestion, I'd scope your variables and create $mechanize on each iteration. This may help the garbage collection to free up the memory

#!/usr/bin/perl 
use WWW::Mechanize;
use HTTP::Cookies; 
use Stream::Reader;

$url="https://mymail.company.com";
my $username = "XXXXXXXXXX"; 
my $password = "xxxxxxxxxx";
CHK_STRT:
{
  my $mechanize = WWW::Mechanize->new(autocheck => 1);
  $mechanize->cookie_jar(HTTP::Cookies->new());
  $mechanize->credentials($username,$password);
  $mechanize->get($url);
  $mechanize->get("https://mymail.company.com/exchange/User/Inbox/?Cmd
+=contents&Page=1");
  my $page = $mechanize->content();
  $mechanize->get("https://mymail.company.com/exchange/User/Inbox/Sub_
+dir_1/?Cmd=contents&Page=1");
  my $Sub_1>get("https://mymail.company.com/exchange/User/Inbox/Sub_di
+r_2/?Cmd=contents&Page=1");
  my $Sub_2 = $mechanize->content();
  $mechanize->get("https://mymail.company.com/exchange/User/Inbox/sub_
+dir_3/?Cmd=contents&Page=1");
  my $Sub_3 = $mechanize->content();
  $mechanize->get("https://mymail.company.com/exchange/User/Inbox/sub_
+dir_4/?Cmd=contents&Page=1");
  my $Sub_4 = $mechanize->content();
  
  open(FH, ">school.txt") or die " Can't open school file\n";
  binmode FH, ':utf8';
  print FH $page;
  print FH $Sub_1;
  print FH $Sub_2;
  print FH $Sub_3;
  print FH $Sub_4;
  close(FH);
  
  
  my @substrings = (
  'icon-msg-unread.gif'
  );
  my $handler;
  open( $handler,'<','school.txt' ) or die "can't Reopen the file\n"; 
  my $stream = Stream::Reader->new( $handler );
  my $result = $stream->readto(\@substrings, {Mode => 'E'}); #This mod
+e returns false 
  $emails = 1;
  close $handler;
  open(WR, ">announce.txt");
  if( $result ) {
  print WR "new\n"; $emails++; 
  } elsif( $stream->{Error} ) {
    die "Fatal error during reading file!\n";
  } else {
      print WR "old\n";
  }
  close WR;
}
unlink('C:/MC/school.txt');
system 'ftp -s:ftpc ftp.server > Log.log';
unlink 'Log.log';
sleep(60);
goto CHK_STRT;
[download]

I haven't tried this code, but it may make a starting point.