I actually do this on a normal basis at work. Here are a few options.
1. Bring in the clips and duplicate the layers in a new composition and stack them on one another so that one is on top of the other.
2. find a part of the clip where you can draw a mask around a good portion of the shirt that is clean with no logo
3. freeze frame that clip, and place it as the top layer, Freeze Frame it by using the time stretch tool.
4. double click on your mask and center the anchor point.
5. Then go back to your composition and move your mask over with the clean shirt sample and scale it up to cover the logo with some safety area to move your mask during keyframing without seeing the logo
6. Now you can either keyframe the mask to cover the logo or take it a step further by
7. creating a null object and place it on the top layer
8. go to your bottom layer clip and track the motion of the logo
9. apply the tracking to the null object
10. line up your play head where you did the freeze frame and placed your mask over your logo
11. parent the freeze framed mask layer to the null object with the tracking data and scrub through and adjust where necessary