It's not possible to have the page refresh once there is a new post. You should use the refresh meta tag instead, to refresh every 30 seconds or so.
This is because, the HTML page is static, and yes, the PHP is dynamic and can check the database, but only if the page refreshes in the first place.
So, in short, no this is not possible, sorry.
As I said, I'd suggest using the refresh meta tag to refresh every few seconds or so.