Editing object that is in multiple frames of video using after effects

The proper technique depends entirely on the shot. After Effects is commonly used to replace things in a shot. If there is camera movement you will need to do some kind of motion tracking, or with some shots, you stabilize the motion, then replace the element, then add the motion to the replacement layer and restore the motion to the stabilized shot. If something like the persons hand moves in front of the thing you want to replace you'll need to do some rotoscoping by either duplicating one of the layers or using a track matte. These are all common tasks that are done in many different compositing apps including After Effects. In every case the technique is basically the same, match movement and add layers. Here's an example where the blank screen on a TV is replaced with some footage. This is a relatively complex problem to solve and the demonstrated technique is one of many different techniques that could have been used to fix this shot:

The tv screen would be the paper, the footage would be the fix for the logo. If you show us the shot you are working on we can suggest the most efficient way for you to create your FVX shot.