No, during boot of an iOS device only a couple of default applications (springboard, wifi, mobile etc., which could be better called services) are started.
I suspect that apps that use system services might take time to register each app in the appropriate places at startup.
This app are already registered. Let's think of using the push notification service (PNS). During startup of the app they are registering (and renew their registration) to iOS. If you manually close one of this apps or reboot the whole device, the apps is closed, not running and doesn't use any system resources (beside their permanent storage).
If one of this apps is notified by the PNS (or the local notification service), you get a iOS (not app specific) notification and if you then open the app, a special (pre-defined) method
(void)application:(UIApplication *)application didReceiveRemoteNotification
is called and the developer can decide how to handle this kind of event. If you dismiss the notification, the app is not executed.
bmike wrote, the springboard load time depends of course on the number of installed apps, but I agree that this is small. Also in this this scenario the apps are actually not executed.
To summarize this:
- Number of apps installed influence the boot time only marginal (see @bmike)
- App which are running at the previous shutdown does not influence the boot time.