Is there any way to set an application, in this case Gwibber, in the background of the desktop?
This means basically having it in the background showing tweets and if I don't click it, like, I don't know, twice, then I'm not "interacting" with it and it is not enabled.
If the question is not clear comment it and I will correct what's unclear.
Through various Compiz plugins (from the extra package, I guess), you can maximize a window (by title), and put it in the background (possibly even making it unclickable). I'm afraid that I forgot what the specific plugins were.
In addition, I never tried this with Unity. I expect results better than before, because the programs menu bar will never be visible, and maximized windows only have controls in the panel -- which will be busy doing other things.