If you've linked your Youtube account to an actual Google account, this confuses the iOS app and prevents you from logging in. Anyway, here's how to fix it.
Login to Google's 2 step verification page: https://accounts.google.com/b/0/SmsAuthConfig
After turning on verification, you want to scroll down and hit: Generate Application specific password.
Create a name for what this password will sync to and hit "Generate". Youtube will create a 4 4-digit codes (not a type, looks like: abcd efgh abcd efgh). Then go back to the Youtube app on your iDevice, log with the gmail address and type in that generated password.
If you're still having problems, check this video: https://www.youtube.com/watch?v=FXW5D17UyH4&feature=g-all-u