Traditionally this is done through a technique called Green Screening you would wear gloves of a given colour and the software would replace anything of that colour with the background from previous shots - but a) you would need to wear the gloves & b) the changes to the screen would not show until you moved your hands out of the way.
You could take an alternative approach by using a product such as Mobizen. This consists of an application on your phone and on your PC which allows you to either screen record on your PC what is happening on your phone or control your phone using your PCs mouse & keyboard while videoing the phones screen. Then of course you have no fingers in the shot to remove. A video of this in action is available here.
Connection to the mobile is via USB, WiFi or Web.