It seems that you need some information about how the Slingbox will help you to control the TiVo boxes at home, when running the SlingPlayer Mobile application on your iPhone.
First, you should get either the Slingbox PRO or the Slingbox PRO HD, since they will allow you to connect more than one TV source.
The following links will allow you to get all the information you must know about the Slingbox PRO and the Slingbox PRO HD features.
The SlingPlayer Mobile application for iPhone will allow you to access and control all of the features of your DVR as long as it is included on the supported devices list. Check these links in order to get all the information about the SlingPlayer Mobile application for the iPhone and supported devices list.
Finally, if your TiVo box is a dual tuner box, you will be able to "split" the TV signal without affecting home viewers. Otherwise, you and the home viewers will be sharing the same TV source.