No. Connections are closed as soon as the script finishes
Using mysql_close() isn't usually necessary, as non-persistent open links are automatically closed at the end of the script's execution.
Persistent connections will not work either
People who aren't thoroughly familiar with the way web servers work and distribute the load may mistake persistent connects for what they're not. In particular, they do not give you an ability to open 'user sessions' on the same link, they do not give you an ability to build up a transaction efficiently, and they don't do a whole lot of other things. In fact, to be extremely clear about the subject, persistent connections don't give you any functionality that wasn't possible with their non-persistent brothers.
Additionally, even if you could somehow reserve a connection, the resources associated with it would not be available
Due to the reference-counting system introduced with PHP 4's Zend Engine, it is automatically detected when a resource is no longer referred to (just like Java). When this is the case, all resources that were in use for this resource are made free by the garbage collector.
Bottom line is that you will have to try another approach.
I should point out that there is some hope for you, the mysql query cache
From version 4.0.1 on, MySQL Server features a query cache. When in use, the query cache stores the text of a SELECT statement together with the corresponding result that was sent to the client. If an identical statement is received later, the server retrieves the results from the query cache rather than parsing and executing the statement again.
So it depends on the nature of your app as to whether the cache will solve your problem.