Instead of using tee, have you considered just printing directly to the file? I assume you are wanting the results to print to screen as well as to a log file. Why not do something like this:

open(FILE, ">$name") or die "Failed file open: $!\n";

my $stmt = $dbh->prepare("$sql") or die $DBI::errstr; # $sql takes 30-
+40 secs. to run
$stmt->execute() or die $DBI::errstr;

while (my @array = $stmt->fetchrow()) {
    print;
    print FILE "$_";
}
$stmt->finish();

close(FILE);
[download]

I didn't test this, but I'd bet that it's faster than running tee -- not to mention quite a bit easer to customize.
Hope that helps,
Shendal