This has changed since iOS 9 with Wi-Fi Assist
With Wi-Fi Assist, you can stay connected to the Internet even if you have a poor Wi-Fi connection. For example, if you're using Safari with a poor Wi-Fi connection and a webpage doesn't load, Wi-Fi Assist will activate and automatically switch to cellular so that the webpage continues to load. You can use Wi-Fi Assist with most apps like Safari, Apple Music, Mail, Maps, and more.
Prior to iOS 9…
No, unfortunately not.
If the WiFi signal itself is still 'live' the iPhone doesn't check to see if it's actually connected to the outside world, unfortunately. It will just spin its wheels waiting for data that never arrives.
You need to switch it over manually.