Other apps and processes may be using data in the background. When you go into the Data Usage menu (again assuming you have ICS), the list will show you which apps and processes are using data and how much.
And if by "data mgr" you are in fact referring to the phone menu, that is a setting within the phone itself and has nothing to do with your actual data plan. You can actually adjust the maximum data allowed before getting a warning and it really doesn't matter unless you set it to correlate with what your plan allows. Even then, what the phone measures as used data may not be the same as what your carrier measures. What really matters is what your service provider allows and whether they tell you that you are exceeding that amount, if you don't get a warning from them I wouldn't worry about it. Unless you have a very low data cap I doubt a week of using TuneIn would put you over.
As for the app itself and not being able to set it for wifi only, I don't think you'll find many (or any) apps that have this option. So don't blame the developer, it's up to the user to know what data connection they are using and to use or not use the app accordingly.
Finally if you want to reduce the amount of data you use when streaming with TuneIn, go to the options and select "Choose Stream". The app may be defaulting to the highest bandwith stream available (you can select or deselect this as an option in the settings), but you can manually select a smaller stream with a minimal reduction in sound quality. I find that using a 48K AAC stream, when one is available, provides almost the same sound quality as a 192K or 128K MP3 while using much less data.