Either remap your UVs with
smart UV project and set
island margin to an appropriate value (Experiment, 0.02 works for me), or manually increase the size of the margin around each island by a few pixels. You can easily do this by hand with a photo-editing tool.
The black lines are from the mipmapping used in realtime graphics pipelines, including Blender's editor and Unity. The mipmapping works fine at full resolution (eg: when close to the model) but when the camera is far away, adjacent colours tend to blend together. On your texture, coloured islands are adjacent to black areas, so you have black lines.
This Gamedev.SE answer is wonderful and explains the mipmap bleed effect very well.