I can access the youtube Playlist with the default youtube application on an iOS device by tapping "More" then "Playlists", but how can I play those lists? When I tap on a playlist, it shows the list of videos included in it, and I can only play them individually one at a time. I want to play the whole list continuously. I don't connect my device (iPod Touch 4) to a computer, so I cannot use a solution that uses iTunes.
There is no way you could do that, I have that problem also. You could try to send YouTube a feature request, but I won't think they will do that as Google now owns YouTube and Google and Apple aren't exactly the best friends... But you coud try :)
Download the google app. Next open the google app on your iphone and type "youtube" into the search bar. Click youtube.com search result to open the youtube website and it will automatically open the mobile version.
Now you can play playlists and use all of youtubes functionality on your iphone.