I don't have GoogleTV, but it would appear that your TV allows you to stream content from the web via GoogleTV integrated apps, such as Netflix, Hulu, Amazon, etc. For the TV to do such streaming, it needs an internet connection. Plug it in and you are able to stream. This is independent of TiVo mostly (ignoring Ethernet over MOCA).
Your TV should allow you to display content from TiVo on (let's say) HDMI1 or content from GoogleTV. Since the TV is controlling the input source, it's no different from watching a DVD or the TiVo. They are both connected, but operate independently. In your case, GoogleTV is not connected, but built-in, however it operates similar to the way your DVD does. Your DVD viewing and GoogleTV viewing has no impact on what TiVo is doing.
My TV offers similar streaming, but not using GoogleTV. In practice, if TiVo offers the same, I would use that instead of the TV. I prefer to stick with one remote and interface. Watched a YouTube video (Reactions to Red Wedding) again last night and never considered using my TV as the provider, nor did I consider using my BluRay player for this either (and it also streams). This is simply a personal choice of how you access that content.
Last edited by bantar; 06-10-2013 at 02:36 PM.