I am having the same problems when reprojecting satellite imagery using the Project Raster Tool.
I have found that the edge pixels (black lines) in the projected imagery display in ArcMap, even when I symbolize the image tiles to Display the background value (0,0,0) as No Color. The problem goes away when you zoom in to scales of < 1:5000 in my imagery, and appears if I am zoomed to scales of >= 1:5000. This led me to believe that the problem is related to the Pyramids that are built automatically when you use the Project Raster tool. Pyramids are used for image display performance. If you remove the Pyramids, the issue goes away and the imagery displays fine - no black lines on the edges of the image tiles. To do this, open your Windows Explorer and look for a .rrd file for each projected image. (For example Image1.rrd, etc) Either delete the .rrd files, or move them to another folder to test my theory. In my case,.I really do not want to delete the Pyramids, because they speed up the display of the imagery. So I need a solution that allows me to keep the Pyramids, but not have the black edge pixel problem