The original thread thread226-1560593: cutting compound paths is closed so I'm restarting. I had a problem where I had to cut several open paths by using a shape. I could not use a mask because the mask simply hides the beyond the mask lines, and I needed to completely remove them (this is going into a laser cutter). The solution I found through random poking around is as follows:

1. Select the lines you want to cut and the object you want to use as the cutting mask. Then go Object - Live Paint - Make, then Object - Live Paint - Expand.
2. You'll now have anchor points whenever the paths cross. Select the object you used as the cutting mask and remove it. Now use the white arrow selection tool and drag around the area where the object was to select all the newly created anchor points.
3. At the top of the window (the control bar) you'll see a button with scissors and a curved path. Click on this button to cut at all selected anchor points.
4. Select all your cut paths and ungroup them. You can now simply select the unneeded path segments and remove them.

Hope this helps.