Yes, the three HTTP monitor types do allow for host headers. A host header allows you to send the appropriate domain name with the IP address being tested. Often this is required if you have multiple websites behind the same IP, or if you have configured your website not to load by IP address only. It is almost always required for shared hosting accounts too because testing just against the IP address often yields only a test or default page that has nothing to do with your website.
To specify the host header you must first configure it as an Advanced Monitor. Then, put the fully qualified domain name into the Custom Header box on the standard parameters tab of the HTTP, HTTP-ECV or HTTP-INLINE monitor. For example, if you need to test against www.example.com, you would put exactly that “www.example.com” (without the quotes) into the custom header box. See the example below.