Just to clarify this note in the Manual regarding this function:
"Note: When using mysqli_stmt_execute(), the mysqli_stmt_fetch() function must be used to fetch the data prior to performing any additional queries."
This is because this function DOES NOT store the result set on the client side so you have to fetch everything in the result set or else you risk major errors.
If you however use the function mysqli_stmt_store_result immediately after you use this function, you are forcing the result set to be stored on the client side and thus it is safe to issue extra queries before fetching all the data.
This is where you really have to make a choice regarding on your application's priorities. If you know your result set is memory hefty, then its a good idea not to store it on the client side so you don't run in any errors regarding unavailable memory on the server. But this also means your not going to do a lot of calculations on the result set or else you will prevent any other usage of the table to which the result set came from until you fetched it all.
If your going to do a lot of calculations or your result set is not memory hefty, its probably a good idea to store it on the client side.
Most of these problems can easily be solved if you have a lot of memory available on your server but thats usually not the case for those on shared hosting.
An intelligent way to counter this problem if your on a shared host is to be smart in the way you design your queries. Try to limit the result set if you know you will be fetching memory hefty result sets.
Test different alternatives for your application and see what works best for you under different conditions.
Good Luck,