The Template Toolkit language isn't really perl. It is primarily about text substitution, not computation. It's possible to embed real perl in a template with the PERL directive. Here's your code with that modification,

#!/usr/bin/perl -w

use strict;
use Template;

my $template=new Template( EVAL_PERL => 1);
$template->process(\*DATA) || die $template->error;

__DATA__
8 and 1 = [% PERL %] print 8 & 1 [% END %] (should be 0, if we use a b
+inary operator)
8 or 1 = [% PERL %] print 8 | 1 [% END %] (should be 9, if we use a bi
+nary operator)
[download]

When run, that produces,

$ perl ttbin.pl
8 and 1 = 0 (should be 0, if we use a binary operator)
8 or 1 = 9 (should be 9, if we use a binary operator)

$
[download]

"Binary operator" usually refers to an op with two arguments, usually with infix semantics. We usually call the operators you wanted "bitwise". In perl, and and or are low-precedence versions of the logical operators && and ||.