I've just discovered something on Illustrator!

No more manual erasing at Photoshop: In Illustrator, highlight all the paths whose bits of lines you want to trim out, then go to Windows>Pathfinders, under the pathfinder window on the pathfinder segment, and click on the Outline icon (second one from right).

Next, double-click on the combined path to open the whole group, click on the area you want to delete, and press the Del key.

Sometimes, it works on Divide (first from left under the pathfinder segment). I think Outline is the right button to use, though.