To see if a process is executed with elevated rights or not: open the Process Explorer (as Administrator), find the process you need to check, then open the property's Security tab.
If the "BUILTIN\Administrators" record says "Deny" - that means the process is standard,
if it says "Owner", then it's elevated.
IIS's pool process (w3wp.exe) is executed elevated, so no need to worry, but need make sure if the application pool is running as "LOCAL SERVICE", that "LOCAL SERVICE" have rights to the resources the web application need access to. Also to give it more rights, in the web application setting check "Same as the application pool".