The only reason to leave connections open is if you are using persistent connections. However, they are another area entirely.
Otherwise, you will want to close connections. Why? Because databases have a maximum number of connections. Therefore is you exceed this you wont be able to connect anymore.
So the connection will close after the script ends but I assume it will be left open for longer than it needs to be. Perhaps until it times out?
So, assuming you are using the mysql_* functions. All you need to do is put mysql_close at the end of the pages that use the db.